fuzzy granule algebraic structures in grc

International Research Journal of Computer Science (IRJCS) ISSN: 2393-9842 Issue 1, Volume 2 (January 2015) www.irjcs.com

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Fuzzy Granule Algebraic Structures in GrC Hongbing Liu, Wenyong Zhou, Haitao Mao

School of Computer and Information Technology, Xinyang Normal University Xinyang 464000, Henan Province, China

Abstract: Operations between two granules and inclusion measure are two important issues in granular computing. Two operations between two granules, named join operation and meet operation, are formed to realize the transformation between granule spaces with different granularities, and the fuzzy inclusion measure is induced by the aforementioned operations. The granule set, operations, and fuzzy inclusion measure, are used to form the fuzzy granule algebraic structures, which are proved as fuzzy lattices. We designed the classification algorithms based on the formed fuzzy granule algebraic structures. The experimental results showed that the feasibility of GrC induced by he formed fuzzy granule algebraic structures 2-dimensional space and N-dimensional space.

Keywords: Granule, join operation, meet operation, algebraic structure

1. INTRODUCTION Granular computing (GrC) concerns the processing of complex information entities called information granules, which arise in the process of data abstraction and derivation of knowledge from information or data [1,2]. In the philosophical sense, granular computing can describe a way of thinking that relies on the human ability to recognize the real world under various levels of granularity in order to abstract and consider only those things that serve a specific interest and to switch among different granularities. By focusing on different levels of granularity, one can obtain different levels of knowledge, as well as a greater understanding of the inherent knowledge structure. Granular computing is thus essential in human problem solving and hence has a very significant impact on the design and implementation of intelligent systems, such as classification problems [3-5].

Classification problem, which is the task of assigning objects to one of several predefined categories, is a pervasive problem that encompasses many diverse applications. The general frame of classification can be described as FIGURE 1. The frame includes three aspects, such as input, classification model, and output. Input is a vector composed of the attribute values, classification model is induced based on a training set of data containing observations (or instances) whose category membership is known, and output is the class label assigned to input. In general, classification problem is to form the relationship between input and output on the training set, the formed relationship is used to predict the class labels of data whose class labels are unknown. The formed relationship can be analytic function, such as SVMs [6], relation between two objects, such as KNN [7], the structural model, such CART[8].

FIGURE 1: The classification process

The input data for a classification task is a collection of records. Each record, also known as an instance or example, is characterized by a tuple (x,y), where x is the attribute set and y is a special attribute, designated as the class label (also known as category or target attribute). Although the attributes are mostly discrete, the attribute set can also contain continuous features. The class label, on the other hand, must be a discrete attribute. This is a key characteristic that distinguishes classification from regression, a predictive modeling task in which y is a continuous attribute. The rest of this paper is presented as follows: Section 2 introduces the motivation and related works. Fuzzy granule algebraic structures are described in Section 3. Section 4 demonstrates the comparative experimental results on classification problems. Section 5 summarizes the contribution of our work.

2. MOTIVATION AND RELATED WORK In this section, the motivation for this proposed research work is presented, and some related works are discussed.

2.1 MOTIVATION In granular computing, two issues, such as the representation of granule and inclusion relation between two granules, must be discussed. From the view of set theory, the granule is represented as the subset of universal U, and the inclusion relation between two granules is the inclusion relation between two set which is crisp. If the granule is represented as the set, the shape of granule is irregular. Granularity (the size of granule) of granule is induced by the distance between any two elements belonging to the granule, and the union and intersection of two sets are irregular. It is difficult to compute the granularity and design the operation between two granules. To study granular computing expediently, the granules are represented as the normal forms with the center and granularity, such as the hyperdiamond granule, the hypersphere granule, and the hypercube granule. The joint operation ∨ and meet operation ∧ between two granules are designed to transformation between granules with different granularity. The fuzzy inclusion measure between two granules is induced by the join operation ∨ and meet operation ∧. The fuzzy granule algebraic structures are formed by the granule set and fuzzy inclusion measure between two granules.

International Research Journal of Computer Science (IRJCS) ISSN: 2393-9842 I s su e 1, V ol u me 2 ( J anu ary 20 15 ) www.i rjc s.c om


2.2 RELATED WORK Granular computing has been proposed and studied in many fields, including machine learning and data analysis [9,10]. Two granular structures induced by a rough set were proposed by Yao: one is a partition induced by an equivalence relationship, and the other is a covering induced by a reflexive relationship. Each equivalence class can be viewed as a granule, and each block induced by the similarity relationship is regarded as a granule. Yao also suggested the inclusion measure to form granular structures. A inclusion measure of two sets is defined as (A,B)=|AB|/|A|, can be interpreted as the conditional probability that a randomly selected element in A belongs to B, which can be used to measure the degree to which A is a subset of B. can be interpreted as a fuzzy partial order relation of 2U, and the use of a complete lattice corresponds to the lattice-based fuzzy partial order relations in the fuzzy set theory. Kaburlasos and colleagues proposed a fundamentally new and inherently hierarchical approach on neurocomputing, called fuzzy lattice neurocomputing (FLN) [5]. Based on FLN, they designed fuzzy lattice reasoning (FLR) classifiers in which the partially ordered relationship is induced by the positive valuation function. FLR classifiers are applied to air-quality assessment [11] and estimation of ambient ozone [3], using both lattice theory and granular computing. The difference between the granule structure proposed by Yao and the granular computing introduced by Kaburlasos is that the fuzzy inclusion measure between two granules in the granular computing is computed by the ratio of the granule to the join of two granules or the ratio of the meet of two granules to the original granule.

3. FUZZY GRANULE ALGEBRAIC STRUCTURES For the data set S={(xi,yi)|i=1,2,...,n} of classification problems in N-dimensional space, we construct fuzzy granule algebraic structures in terms of the following steps. Firstly, a granule is represented as a subset of S which is composed by the data with the same class label, and the size of granule is measured by the granularity induced by the maximal distance between data belonging to the same granule. Secondly, operations between two granules are designed by the two granules and their granularities, and the fuzzy inclusion measure between two granules is induced by granule and the operated granule. Thirdly, the algebraic structures are formed by granule sets, inclusion measures between two granules, and operations between two granules, and the formed algebraic structures are proved as lattices or fuzzy lattices. Finally, the formed algebraic structures is used to guide the designs of classifiers.

3.1 REPRESENTATION OF GRANULE In reality, the shapes of granules are irregular. In order to study granule, the shapes are represented as the regular forms. In 2-dimensional space, the granule has the form of diamond, circle, and cube. In general, the attributes of input are more than 2, namely, the classification problems are performed in N-dimensional space, and the granule is represented as some forms in N-dimensional space. A granule is represented as the form of vector G=(C,R), where C is the center of granule, R is the size of granule, and refers to the granularity of granule G. Particularly, a point x is represented by a atomic granule with the central vector x and granularity 0 in N-dimensional space. Different forms of distances denote different shapes of granules. The distances between center C=(c1,c2,...,cN) and datum x=(x1,x2,...,xN) can be defined as follows d1(x,C)=|x1-c1|+|x2-c2|+...+|xN-cN| d2(x,C)=((x1-c1)2+(x2-c2)2+...+(xN-cN)2)1/2

d3(x,C)=max{|x1-c1|,|x2-c2|,...,|xN-cN|} The granules have the forms of hyperdiamond, hypersphere, and hypercube in N-dimensional space. FIGURE 2 shows three shapes of granule G=(1,2,1). The center is (1,2), the granularity is 1, d1(,) induces the diamond granule (hyperdiamond granule in N-dimensional space), d2(,) induces the circle granule (hypersphere granule in N-dimensional space), and d3(,) induces the cube granule (hypercube granule in N-dimensional space).

FIGURE 2: The granules with different shapes

3.2 OPERATIONS BETWEEN TWO GRANULES The operations between two granules reflect the transformation between macroscopic and microcosmic of human cognitions. When a person want to observe the object more carefully, the object is partitioned into some suitable sub-objects, namely the universe is transformed into some parts in order to be studied in detail in the view of microscopic. Conversely, there is the same attributes of some objects, we study the objects as a universe to simple the process in the view of macroscopic. The operations between two granules are designed to realize the transformation between macroscopic and microscopic.



Set-based models of granular structures are special cases of lattice-based models, where the lattice join operation ∨ coincides with set union operation ∪ and lattice meet operation ∧ coincides with set intersection operation ∩. Join operation ∨ and meet operation ∧ are used to realize the transformation between macroscopic and microcosmic. Operation ∨ unites the granules with small granularities to the granules with the large granularities. Inversely, Operation ∧ divides the granules with large granularities into the granules with small granularities. Join operation ∨ and meet operation ∧ are designed as follows. Any points are regarded as atomic granules which are indivisible, the join process is the key to obtain the larger granules compared with atomic granules. Likewise, the whole space is a granule with the maximal granularity, the meet process produces the smaller granules compared with original granules. For two granules G1=(C1, R1) and G2=(C2, R2) in N-dimensional space, the central vector C of G and the granularity R of the join granule G=G1G2=(C, R) are computed by algorithm1.

Algorithm1. computing C and R of join granule G between G1 and G2 Input: G1=(C1,R1) and G2=(C2,R2) Output: G=(C,R) if C1=C2 if R1>=R2

C=C1

R=R1 else C=C2

R=R2 end else C12=(C2-C1)/d(C1,C2) C21=(C2-C1)/d(C2,C1) P1 = C1-C12R1 Q1 = C2-R2C21 if R1>=R2 if d(C1,C2)<=R1-R2 C=C1

R=R1 else C=(P1+Q1)/2 R=d(P1,Q1)/2 end else if d(C1,C2)<=R2-R1 C=C2

R=R2 else C=(P1+Q1)/2 R=d(P1,Q1)/2 end end end

Similarly, the center c and granularity r of the meet granule g=G1G2=(c,r) is computed by algorithm2. Algorithm2. computing c and r of the meet granule g between G1 and G2 Input: G1=(C1,R1) and G2=(C2,R2) Output: g=(c,r) if C1=C2 if R1>=R2

C=C2

R=R2 else C=C1

R=R1 end else C12=(C2-C1)/d(C1,C2) C21=(C2-C1)/d(C2,C1) P2 = C1+C12R1 Q2 = C2+R2C21



if d(C1,C2)>R1+R2 c=(P2+Q2)/2 r=-d(P2,Q2)/2 else if R1>=R2 if d(C1,C2)<=R1-R2 c=C2

r=R2 else c=(P2+Q2)/2 r=d(P2,Q2)/2 end else if d(C1,C2)<=R2-R1 c=C1

r=R1 else c=(P2+Q2)/2 r=d(P2,Q2)/2 end end end end

For algorithm1 and algorithm2, if d(,)=d1(,), the operations are induced by two hyperdiamond granules, if d(,)=d2(,), the operations are induced by two hypersphere granules, if d(,)=d3(,), the operations are induced by two hypercube granules. FIGURE 3 shows the join process of the hyperdiamond granule G1 = [0.2 0.15 0.1] and the hyperdiamond granule G2 = [0.1 0.2 0.1]. The crosspoints of hyperdiamond granule G1 and the line crossing vector C12=[-0.6667,0.3333] are P1=[0.2667, 0.1167] and P2=[0.1333,0.1833]. The crosspoints of hyperdiamond granule G2 and the line crossing vector C21=[0.3333 -0.6667] are Q1=[0.0333,0.23333] and Q2=[0.1667,0.1667]. According to algorithm1, the central vector and granularity of the join hyperdiamond granule G are C=[0.15,0.175] and R=0.175, namely G=[0.15 0.175 0.175]. Similarly, the meet hyperdiamond granule is g=[0.15, 0.175, 0.025] shown in FIGURE 4.

FIGURE 3. The join hyperdiamond granule of two hyperdiamond granules

FIGURE 4. The meet hyperdiamond granule of two hyperdiamond granules



For hypersphere granule G1 = [0.2 0.15 0.1] and hypersphere granule G2 = [0.1 0.2 0.05], the join hypersphere granule is G=[0.1724, 0.1638, 0.1309] shown in FIGURE 5, and the meet hypersphere granule is g=[0.1276, 0.1862, 0.0191] shown in FIGURE 6.

FIGURE 5: The join hypersphere granule of two hyperspere granules

FIGURE 6: The meet hypersphere granule of two hypersphere granules

For hypercube granule G1 = [0.2 0.15 0.1] and hypercube granule G2 = [0.08 0.25 0.06], the join hypercube granule is G=[ 0.16, 0.1833, 0.14] shown in FIGURE 7, and the meet hypercube granule is g=[0.12, 0.2167, 0.02] shown in FIGURE 8.

FIGURE 7: The join hypercube granule of two hypercube granules



FIGURE 8: The meet hypercube granule of two hypercube granules

As mentioned above, for all G1,G2GS, G1G1G2 and G2G1G2, G1G2G1 and G1G2G2. Namely, the operations between granule G1 and granule G2 are corresponding to the inclusion relation between granule G1 and G2.

G1G2G1G2=G2, G1G2=G1 (1) The inclusion relation between two granules is induced by the operations between two granules.

3.3 FUZZY INCLUSION MEASURE As a subset of training set S, the inclusion relation is fuzzy. For example, G1=[0.1,0.2,0.1] and G2=[0.32,0.2,0.1] have the same granularity and different central vector, and G=[0.4,0.3,0.06]. The inclusion measure between G1 and G are different from the inclusion measure between G2 and G, especially there is overlap between G2 and G. The fuzzy inclusion measure is used to measure the fuzzy inclusion relation. In FIGURE 9, one hand is the difference of the join hypercube granule G1∨G=[0.23, 0.2433, 0.23] and the join hypercube granule G2∨G=[0.344, 0.23, 0.13], the other hand is the difference of the meet hypercube granule G1∧G= and the meet hypercube granule G2∧G=[0.367, 0.27, 0.03]. The fuzzy inclusion measure between G2 and G is greater than the fuzzy inclusion measure between G1 and G.

FIGURE 9: The different inclusion measure between two granules

The join granule and the meet granule are used to measure the fuzzy inclusion relation. The granularity R is used to define the fuzzy inclusion measure.

K(G1,G)=v(G)/v(G1∨G) (2a) S(G1,G)=v(G1∧G)/v(G) (2b)

Where v(G) is the positive valuation function defined by V G kurblasos, which can be the linear function or nonlinear function [3-5]. A valuation function v: L → R is defined on a lattice L as a real function that satisfies: v(a)+v(b)=v(a∧b)+v(a∨b), a,b∈L. A valuation function, is called positive if and only if a<b⇒v(a)<v(b) [12]. Theorem1. For G=(C,R), v(G)=R+, where is a constant, is a positive valuation function defined on GS. Proof. For classification problem S, the granule set and operations and induce the algebraic structure GS,, which is a lattice with the greatest lower bounds and the least upper bounds S. We prove v(G) is a positive valuation function from the following two properties. Firstly, we prove the property G1G2⇒v(G1)<v(G2) as follows. Suppose G1=(C1,R) G2=(C2,R2), if G1G2, then R1<R2, namely, v(G1)<v(G2) Secondly, we prove the property v(G1)+v(G2)=v(G1∧G2)+v(G1∨G2) as follows. If there is a margin between G1 and G2 (see FIGURE 10), then the meet granule does not exist and is called as the pseudo granule, v(G1G2)=-r+, where r is the granularity of pseudo meet granule g=G1G2. v(G1G2)=d1(P1,Q1)/2+=(d1(C1,C2)+R1+R2)/2+



v(G1G2)=-d1(P2,Q2)/2+=-(d1(C1,C2)-R1-R2)/2+ v(G1G2)+ v(G1G2)=R1+R2+2=v(G1)+v(G2)

FIGURE 10: The margin between two hyperdiamond granules

If there is overlap between G1 and G2 (see FIGURE 11), v(G1G2)=d1(P1,Q1)/2+=(d1(C1,C2)+R1+R2)/2+ v(G1G2)=d1(P2,Q2)/2+=(d1(C1,C2)-(d1(C1,C2)-R1)-(d1(C1,C2)-R2))/2+ =(R1+R2-d1(C1,C2))/2+ v(G1G2)+v(G1G2)=R1+R2+2=v(G1)+v(G2)

FIGURE 11: The overlap between two hyperdiamond granules

If G1 is included in G2, then G1G2=G2, G1G2=G1 v(G1G2)+v(G1G2)=v(G1)+v(G2) (3)

3.4 FUZZY ALGEBRAIC STRUCTURES In recent years, lattice computing is used in classification problems[13-16]. For a training set S={(xi,yi)|i=1,2,…,n} of classification problem, every datum xi is represented as an atomic granule which is indivisible. The granule set GS is induced by the representation method of granule, the join operation and meet operation are two operations on granule set GS, the fuzzy inclusion measures are defined by formula (2). So the fuzzy algebraic structures are formed by the granule set induced by representation of granule, operations between two granules, and fuzzy inclusion measure between two granules. A fuzzy lattice is a pair〈L,µ〉, where L is a lattice and (L×L,µ) is a fuzzy set with membership function µ: L×L→[0,1] such that µ(a,b) = 1 if and only if a≤b. If L is a set, the membership is an inclusion measure on L, which is defined as a real function µ: L×L→[0,1], such that for each a,b,x∈L the following four conditions are satisfied [12]. (1)µ(a,) = 0, a≠ (2)µ(a,a) = 1, ∀aL (3)a ≤ b ⇒µ(x,a) ≤µ(x,b) (4)a∧b < a ⇒µ(a,b) < 1. Theorem 2. Algebraic structure〈GS, K(.,.)〉is fuzzy lattice, where K(G1,G2)=v(G2)/v(G1∨ G2), v(G)=R+. Proof. Firstly, we prove GS is lattice for operations ∨ and ∧ as follows. Because the inclusion relation is a partially ordered relation, 〈GS,∨,∧〉is lattice with the greatest lower bounds and the least upper bounds GS. Secondly, we prove K(.,.) satisfies the aforementioned four condition of an inclusion measure on GS.



(1) v()=0, K(G, )=v()/v(G∨)=0 (2) GG=G, v(G)/v(G∨G)=1, namely K(G,G)=1 (3) if G1G2, then GG1GG2 and v(G1)v(G2). GG1GG2v(GG1)≤v(GG2), G1 GG1v(G1)v(GG1), G2GG2v(G2)v(GG2), v(G1)/v(GG1)v(G2)/v(GG2), namely K(G, G1) K(G,G2). (4) G1∧G2 <G1, v(G1∧G2)<v(G1), v(G1∨G2)=v(G1)+v(G2)-v(G1∧G2)> v(G1)+v(G2)-v(G1) = v(G2), namely v(G2)/ v(G1∨G2)<1, k (G1,G2)<1. For training set TS, the granular computing classification algorithms are proposed by the following steps. Firstly, the samples are used to form the atomic granule. Secondly, the threshold of granularity is introduced to conditionally union the atomic granules by the aforementioned join operation, and the granule set is composed of all the join granules. Thirdly, if all atomic granules are included in the granules of GS, the join process is terminated, otherwise, the second process is continued. The algorithms include training process and testing process which are listed as follows. Suppose the atomic granules with the same class labels induced by TS are g1, g2, g3, g4, g5. The training process can be described as the following tree structure shown in FIGURE 12, leafs denote the atomic granules, root denotes GS including its child nodes G2 and G3, G1 is induced by join operation of child nodes g1 and g2, G2 is the join granule of G1 and g3, G3 is the join granule of g4 and g5. The whole process of obtaining GS is the bottle up process.

FIGURE 12: The training process of TS including 5 samples

The training algorithm and testing algorithm are described as algorithm3 and algorithm4. Algorithm3. Training process Input: Training set TS, threshold of granularity, the class number n Output: Granule set GS, the class label lab S1. initialize the granule set GS=, lab= S2. i=1 S3. select the samples with class i, and form set X S31. initialize the granule set GSt= S32. j=1 S33. for the jth sample xj in X, form the corresponding atomic granule Gj S34. k=1 S35. compute the distance djk between the atomic granule Gj and the kth granule Gk in GSt S36. k=k+1 S37. find the minimal distance djm S38. if the granularity of the join of Gj and Gm is less than or equal to , the granule Gm is replace by the join, otherwise Gj is the new member of GSt. S39. remove xj until X is empty. S4. GS=GSGSt, lab=lab{i} S5. if i=n, output GS and class lab, otherwise i=i+1 Algorithm4. Testing process Input: inputs of unknown datum x, granule set GS, the class label lab Output: class label of x S1. x is represented as granule g S2. for i = 1:|GS| S3. compute the distance di between g and gi in GS S4. find the minimal distance dm S5. find the corresponding class label of the gm as the label of x

4. EXPERIMENTAL CLASSIFICATION RESULTS AND ANALYSIS We evaluated the effectiveness of our algorithms induced by the fuzzy granule algebraic systems for both two-class problems and multi-class problems, with an Intel Core i5 PC with 3.2 GHz CPU and 8 GB memory, running Microsoft Win7 and Matlab R2008a.

4.1 CLASSIFICATION PROBLEMS IN TWO-DIMENSIONAL SPACE Three spiral curve classification in reference [17] and the classification problem named D31 in reference [18] are used to verify



the feasibility of GrC classification algorithms induced by the formed fuzzy granule algebraic structure. Three spiral data set includes 312 3-class data, including 101 data with class label 1, 105 data with class label 2, and 106 data with class label 3. FIGURE 13 showed the distribution of the data. The positive valuation function v(G)=R+1 for the granule G=(C,R), the threshold is set from 5 to 0 with step 0.01, the maximal classification accuracy and the minimal size of GS are two indicators of selection of parameter . 83 diamond granules (shown in FIGURE 14) classified the 312 data exactly when =3.97, 71 sphere granules (shown in FIGURE 15) classified the 312 data exactly when =3.4, and 67 cube granules (shown in FIGURE 16) classified the 312 data exactly when =3.0. The data set D31 includes 3100 data belonging to 31 classes. We divided the data set into the training set including 2480 data and the testing set including 620 data. The training set is used to form the classification algorithms based on fuzzy granule algebraic systems, and the testing set is used to verify the formed classification algorithms. The threshold ߩ of granularity is from 5 to 0 with step 0.1, the maximal testing accuracy is the selection indicator of optimization algorithms. Performances of GrC with three kinds of shape are listed in Table 1. From the table, we saw that GrC with diamond granules and sphere granules achieved the same testing accuracy, GrC with diamond granule achieved 301 diamond granules when =2.1, GrC with sphere granule achieved 195 sphere granules when =1.8, and the GrC with diamond granule is the optimization algorithm for the minimal size of GS and the minimal training time.

0 5 10 15 20 25 30 350

5

10

15

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25

30

35

data1data2data3

FIGURE 13: Distribution of three spiral curve

0 5 10 15 20 25 30 35-5

0

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35

FIGURE 14: 83 diamond granules induced by GrC in space R2

0 5 10 15 20 25 30 350

5

10

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FIGURE 15: 71 sphere granules induced by GrC in space R2



0 5 10 15 20 25 30 350

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FIGURE 16: 67 cube granules induced by GrC in space R2

TABLE 1: Performance of classification algorithms induced by the fuzzy granule algebraic structure for data set D31 Shape rho Size Training accuracy Testing accuracy Training time Diamond 2.1 301 98.1452 98.0645 1.9968 Sphere 1.8 195 97.9032 98.0645 1.3416 Cube 0.3 1009 99.5161 97.5806 8.3461

4.2 CLASSIFICATION PROBLEMS IN N-DIMENSIONAL SPACE In order to extend the fuzzy granule algebraic structure to N-dimensional space, the traditional image segmentation problems are used to verify the performance of GrC induced by the formed fuzzy granule algebraic structure. The result of image segmentation is a set of segments that collectively cover the entire image, or a set of contours extracted from the image. Olympic banner of five different colors Circle (on behalf of the European sky blue, yellow behalf of the Asian, black Africa, Australia, grass green, red behalf of the Americas) linked to the symbol of unity of the five continents. The aim of this paper is to extract the contours of five different color circles by GrC, because of six kinds of color, the suitable parameter is selected to obtain six granules. The lost of entropy (En) is used to evaluate the segmentations [19], the segmentation with the less En is the better segmentation for the same visual effects. FIGURE 17 are the segmentations by GrC with hyperdiamond granules, hypersphere granules and hypercube granules and their En. According to the En, segmentation by hypersphere granule is the best segmentation because En is minimal.

FIGURE 17: The segmentations by GrC. (a) original image, (b) segmentation by hyperdiamond granules (=0.75), (c) segmentation by hypersphere granules (=0.5), (d) segmentation by hypercube granules (=0.45).

5. CONCLUSION The representation of granule, the join operation and meet operation between two granules, the fuzzy inclusion measures between two granules are discussed in details. The fuzzy granule algebraic structures are induced by the granule set, operations between two granules, and fuzzy inclusion measure between two granules, and used to form the GrC. The experimental results showed the feasibility of GrC which achieved the better generalization ability and extended to N-dimensional space.

CONFLICT OF INTERESTS The authors declare that there is no conflict of interests regarding the publication of this paper.

ACKNOWLEDGMENTS This work was supported in part by the Natural Science Foundation of China (Grant no. 61170202, 61402393)



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