DSpace at VNU: Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization

G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 Applied Soft Computing xxx (2014) xxx–xxx Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization Le Hoang Son ∗ VNU University of Science, Vietnam National University, Viet Nam a r t i c l e i n f o Article history: Received in revised form 14 February 2014 Available online xxx Keywords: Context clustering Fuzzy clustering type-2 Geo-demographic analysis Heuristic algorithms Particle swarm optimization a b s t r a c t Geo-Demographic Analysis, which is one of the most interesting inter-disciplinary research topics between Geographic Information Systems and Data Mining, plays a very important role in policies decision, population migration and services distribution Among some soft computing methods used for this problem, clustering is the most popular one because it has many advantages in comparison with the rests such as the fast processing time, the quality of results and the used memory space Nonetheless, the state-of-the-art clustering algorithm namely FGWC has low clustering quality since it was constructed on the basis of traditional fuzzy sets In this paper, we will present a novel interval type-2 fuzzy clustering algorithm deployed in an extension of the traditional fuzzy sets namely Interval Type-2 Fuzzy Sets to enhance the clustering quality of FGWC Some additional techniques such as the interval context variable, Particle Swarm Optimization and the parallel computing are attached to speed up the algorithm The experimental evaluation through various case studies shows that the proposed method obtains better clustering quality than some best-known ones © 2014 Elsevier B.V All rights reserved Introduction Geo-Demographic Analysis (GDA), which was defined as “the analysis of spatially referenced geo-demographic and lifestyle data”[33], is one of the most interesting inter-disciplinary research topics between Geographic Information Systems and Data Mining, and is widely used in the public and private sectors for the planning and provision of products and services There are various examples showing the needs of GDA in practical applications Shelton et al [34] performed a geo-demographic classification for mortality patterns in Britain and found the main causes of deaths in England and Wales from 1981 to 2000 associated with geographical locations in a map so that they could assist decision makers in better understanding the distribution of major causes Michael [23] conducted a GDA analysis to gather community attitudes on the future growth of Werri Beach and Gerringong, NSW (Nelson), Australia focusing primarily on what actions Council should take to manage population growth within existing neighborhoods Páez et al [29] presented a geo-demographic framework using data from Montreal, Canada to identify potential commercial partnerships that could exploit the characteristics of smart cards Campbell et al [8] ∗ Correspondence to: 334 Nguyen Trai, Thanh Xuan, Hanoi 010000, Viet Nam Tel.: +84 904171284; fax: +84 0438623938 E-mail addresses: sonlh@vnu.edu.vn, chinhson2002@gmail.com provided a detailed GDA of over 37,000 gifted and talented students admitted to the National Academy for Gifted and Talented Youth in England in 2003/2005 and showed that National Academy had nonetheless reached significant numbers of students in the poorest areas, something over 3000 students, and 8% of students identified as gifted and talented at this stage Day et al [11] took a survey that determined clusters of nations grouped by health outcomes by comparing life expectancy and a range of health system indicators within and between each cluster in order to provide sensible groupings for international comparisons Some other typical applications of GDA such as the spatial and socio-economic determinants of tuberculosis, urban green space accessibility for different ethnic and religious groups, children disorders investigation, etc could be referenced in the articles [1,6,9,32,36,37] In order to perform GDA, some soft computing methods are often used such as Principal Component Analysis (PCA), SelfOrganizing Maps (SOM) and clustering Walford [41] described a method using PCA to study the spatial distribution of the 1991 census data scores However, results of PCA depend on the scaling of the variables, and its applicability is limited by certain assumptions made in the derivation Loureiro et al [21] introduced the use of SOM as an adequate tool for GDA Based on the variations in edge length in a path between two units on the SOM, the authors presented a new way of calculating fuzzy memberships of fuzzy clustering method However, it requires a lot of memory spaces to store all neurons and weights; what is more the speed of training http://dx.doi.org/10.1016/j.asoc.2014.04.025 1568-4946/© 2014 Elsevier B.V All rights reserved Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx phase is quite slow Because of some limitations in those methods, clustering is often used instead because it has many advantages in comparison with the rests such as the fast processing time, the quality of results and the used memory space Our previous work in [36] made an overview about some clustering methods for GDA such as Fuzzy C-Mean (FCM) [3], the agglomerative hierarchical clustering [11], Neighborhood Effects (NE) [13], K-Means clustering [20] and Fuzzy Geographically Weighted Clustering (FGWC) [24] Among them, FGWC was considered the most favorite algorithm and was used in most of research articles about GDA applications u k = ˛ × uk + ˇ × × A c wkj × uj (1) j=1 ˛+ˇ =1 wkj = (2) (popk × popj )b a dkj (3) FGWC calculates the influence of one area upon another by Eqs (1)–(3) where uk (uk ) is the new (old) cluster membership of the area k Two parameters ˛ and ˇ are the scaling variables popk , popj are the populations of areas k and j, respectively The number dkj is the distance between k and j Two numbers a and b are user definable parameters A is a factor to scale the “sum” term and is calculated across all clusters, ensuring that the sum of the memberships for a given area for all clusters is equal to one Although FGWC is the most popular clustering algorithm for GDA, it still contains some limitations such as the speed of computing and the clustering quality One of our previous works in [35] presented a method so-called CFGWC to accelerate the speed of computing of FGWC by attaching the context variable terms Other works in [36,37] have showed some preliminary results in improving the clustering quality of FGWC through intuitionistic fuzzy sets and geographical spatial effects Thus, our focus in this work is to continue with the clustering quality problem of FGWC Based upon the observation that FGWC was constructed on the basis of the traditional fuzzy sets, which contain some limitations in membership degrees as pointed out by Mendel [25], this fosters us to improve FGWC in an extension of the traditional fuzzy sets to enhance the clustering quality of the algorithm Now, let us explain why clustering algorithms on the traditional fuzzy sets have low clustering quality According to Mendel [25], the traditional fuzzy sets cannot process some exceptional cases where the membership degrees are not the crisp values but the fuzzy ones instead For example, the possibility to get tuberculosis disease of a patient concluded by a doctor is from 60 to 80 percents after examining all symptoms Even if some modern medical machines are provided, the doctor cannot give an exact number of that possibility This shows the fact that crisp membership values cannot model some situations in the real world and should be replaced with the fuzzy ones Rhee [30] stated that using the traditional fuzzy sets often results in bad clustering quality because their uncertainties such as distance measure, fuzzifier, centers, prototype and initialization of prototype parameters can create imperfect representations of the pattern sets For example, in case of pattern sets that contain clusters of different volume or density, it is possible that patterns staying on the left side of a cluster may contribute more for the other rather than this cluster so that choosing suitable value for the fuzzifier is difficult Bad selection can yield undesirable clustering results for pattern sets that include noise Because of those limitations, some preliminary results of deploying fuzzy clustering methods in an extension of the traditional fuzzy sets so-called Interval Type-2 Fuzzy Sets (IT2FS) have been introduced Mendel [25] described the definition of IT2FS as follows ˜= A (x, u, ˜ (x, u) A = 1)|∀x ∈ A, ∀u ∈ JX ⊆ [0, 1] (4) From Eq (4), we recognize that IT2FS is a generalization of the traditional fuzzy sets since IT2FS will return to the traditional fuzzy sets when there is no uncertainty in the third dimension Based upon this definition, some authors introduced several interval type2 fuzzy clustering algorithms such as in the works of Hwang and Rhee [15] and Rhee [30] Specifically, Hwang and Rhee [15] presented a type-2 fuzzy clustering algorithm to solve the problem of choosing distance measures in FCM algorithm, taking the difference of each type-2 membership function area with the corresponding type-1 membership value Rhee [30] presented an improvement of this algorithm using two different values of fuzzifiers to solve the uncertainty of fuzzifier in FCM Some other variants of the interval type-2 fuzzy clustering algorithms could be referenced in [2,10,12,14,17,19,22,26,27,31,42] Motivated by those results, in this article, we will present a novel interval type-2 fuzzy clustering algorithm so-called Context Fuzzy Geographically Weighted Clustering on IT2FS or in short CFGWC2 to enhance the clustering quality of FGWC The difference of CFGWC2 with those interval type-2 fuzzy clustering algorithms above is two fold: Firstly, CFGWC2 is specially designed for the GDA problem that requires the modification of geographical spatial effects to the algorithm itself; secondly, it is equipped with some additional techniques to speed up the whole algorithm, namely: • An interval context variable, which is an extension of the single context variable of Pedrycz [28], is proposed and used to clarify the clustering results and accelerate the computing speed • In order to avoid bad initialization, which may occur in other interval type-2 fuzzy clustering algorithms, and to converge quickly to the (sub-) optima solutions, a meta-heuristic optimization method namely Particle Swarm Optimization – PSO [18] is used to determine good initial centers for CFGWC2 • Since context values in the interval context variable can be simultaneously processed in CFGWC2, parallel computing technique is adapted to CFGWC2 to reduce the computational costs What have been listed in those bullets are our contributions in this paper The proposed algorithm will be implemented and compared with some relevant methods in term of clustering quality to verify its efficiency The rests of this paper are organized as follows Section “The proposed methodology” elaborates the proposed method in details including those additional techniques one-after-another The numerical experiments through various case studies and discussions are given in Section “Results” Finally, Section “Conclusions” gives the conclusions and outlines future works of this article The proposed methodology In the previous section, we have known that CFGWC2 is an interval type-2 fuzzy clustering algorithm equipped with some additional techniques such as the interval context variable, PSO and the parallel computing for the GDA problem Since those techniques are necessary for the description of CFGWC2, they are firstly presented in Sections “Using PSO for the determination of initial centers” and “The interval context” The CFGWC2 algorithm accompanied with the parallel computing mechanism will be described in Section “Evaluation by various case studies” Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Using PSO for the determination of initial centers point to the ith cluster, for instance, using the sum operator (18) or maximum operator (19) This section mentions the technique that finds good initial centers for clustering algorithms by PSO The idea of this technique is to give a preliminary classification of the original pattern set so that “temporal” cluster results can be used to orient the classification in the main algorithm The objective function is shown in Eq (5), and its constrains are given in Eqs (6)–(7): N C J= Xk − Vj → (5) k=1 j=1 j = 1, C Vi − Vj > max Xs − Vi s=1,POP(i) j= / i Xs ∈ Cluster(i) (6) i = 1, C Cluster(i) ≤ ε1 where POP(i) = and i = 1, C (7) Constrain (6) requires that all clusters are separated from the others Alternatively, the minimal distance from a cluster’s center to the others is not shorter than the maximal one from this center to all data points in the cluster POP(i) is the population or number of patterns in the cluster Cluster(i) Constrain (7) minimizes the number of outliers in the result Accordingly, the number of outliers is not greater than a pre-defined threshold ε1 For the problem (5)–(7), we use PSO [18] to determine the (sub) optima solutions with the beginning population being initiated with P particles Each particle is a vector z = (z1 , z2 , , zC ) where zi (i = 1, C) is a pattern randomly chosen from the original pattern set The velocities of zi are set to zeros Details of the algorithm are described by the pseudo-code in Table Notice that Eq (9) is used solely for the first iteration of MaxStep PSO In the next iterations, the centers are calculated from the previous one Additionally, the value of MDi in Eq (10) is set to zero in case that this cluster has not got any element The fitness value of a particle is calculated by Eq (13) where ( , ) are the ratio constants Eqs (14)–(16) are used to update the velocities and positions of all particles In those equations, c1 is the ratio to keep the velocity intact, c2 is the ratio to change the velocity following by pBest and c3 shows the influence level of gBest to the velocity Since the role of zi (i = 1, C) from the second iteration afterwards is replaced with center Vi , the domain of random number in Eq (14) is set to (−1, 1) in order to ensure the values of the centers are bounded within the domain of the pattern set After a number of iteration steps defined by MaxStep PSO, the solution is getting better because of the amelioration process after each “flying step” based on the fitness function The outputted result V(0) = (V1 , V2 , , VC ) can be found from the particle holding current gBest and is used as the initial center for CFGWC2 The interval context In order to clarify the clustering results and accelerate the computing speed of the clustering algorithms, the context variable could be used According to Pedrycz [28], a (single) context variable in Y ⊂ X is defined through the map below A : Y → [0, 1] yk → fk = A(yk ), (17) where fk can be understood as the representation for the level of relation of the kth point to the supposed context fk There are some ways to define the relation between fk and the membership of kth c uki = fk , k = 1, N, (18) maxuki = fk , k = 1, N (19) i=1 c i=1 In our previous work in [35], we defined a context variable to narrow the original geographical dataset under some conditions of certain dimensions The reason to use the term of context for the clustering algorithm is twofold Firstly, a context variable is useful to clarify the results following by users’ purposes Because only a subset of the original dataset which has considerable meaning to the context is invoked, the result focuses on the area that really has many relevant points Secondly, it helps improving the speed of computing In the traditional clustering method, it not only takes long time to process the whole data, but also makes the results less meaning to the considered context On the contrary, the contextbased clustering methods both accelerate the speed and improve the semantic Nevertheless, there are some limitations in definition (17) Firstly, the importance of the kth point to the supposed context is decided by a value fk In fact, it is not enough to reflect a variety of different evaluations of many people to this relation In the other words, one can assume that the importance is only 0.3 while other affirms that it should be 0.6 Due to this fact, the use of a value fk is not enough Secondly, the old approach excludes the roles of other data points to the context It is a misleading assumption since all characteristics always have relationships either directly or indirectly with the others From these limitations, we extend the use of context by introducing a new term: “the interval context variable” An interval context is defined as f = [f1 ,f2 ] where each fi (i = 1,2) is stated through the map in Eq (17) For the most important points, the value of f is high, e.g [0.6,0.8] Similarly, the value of f in case of less important points is low, e.g [0,0.15] This interval reflects the “fuzziness” of the context In the other words, we have just performed a “fuzzy” step for the considered context It helps us overcome the shortcomings of the single context variable and is suitable for CFGWC2, which works on IT2FS Details of applying the interval context variable for CFGWC2 will be presented in the Section “The CFGWC2 algorithm” The CFGWC2 algorithm We have had a general background of choosing initial centers by PSO in Section “Using PSO for the determination of initial centers” and the basic definition of the interval context in Section “The interval context” Now, we use both of them accompanied with the parallel computing mechanism in the main activity of the CFGWC2 algorithm Let us see the mechanism of CFGWC2 illustrated by Fig below According to Fig 1, the parallel computing mechanism of CFGWC2 requires three machines whose first one (Machine 1) is responsible for generating initial centers for the remaining machines Nevertheless, the centers values of Machine and Machine are different since the stopping conditions of PSO are not identical After (MaxStep PSO/2) iteration steps, the first center V(0) is outputted and transferred to Machine 2, and the second center is sent to Machine after (MaxStep PSO) iterations This guarantees different results in Machine and Machine 3, and is suitable for the determination of the upper and lower centers and membership degrees of the clustering algorithms on IT2FS, i.e U(1) , V(1) (Machine 2) and U(2) , V(2) (Machine 3) in Fig In Machine and Machine 3, we send the initial centers V(0) to a type-2 fuzzy clustering procedure accompanied with the interval Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Table The pseudo-code of PSO procedure Input - The pattern set X whose dimension is r - The number of elements (clusters) – N(C) - The number of particles in the beginning population – P - Maximal number of iteration steps in PSO – MaxStep PSO - Final center V(0) Output Particle Swarm Optimization (PSO) 1: 2: 3: 4: Initialization Repeat For each particle Assign remaining patterns to its clusters: Xj ∈ Cluster(i) ⇔ zi − Xj = zk − Xj |k = 1, C (8) 5: 6: Calculate population POP(i)from current clusters Calculate center Vi and the maximal distance from Vi to cluster’s elements: (l) Vi (l) = Xs /POP(i), l = 1, r, MDi = Xs − Vi max i = 1, C ⎧ ⎨ Xs∈Cluster(i) = s=1,POP(i) max s=1,POP(i) (9) r (Xs (l) − Vi (l) ) ⎩ l=1 ⎫ ⎬ ⎭ , (10) Xs ∈ Cluster(i), 7: Calculate the separated status and the number of outliers: j = 1, C SEP(z) = Cluster(i) OUT (z) = Cluster(i) where Vi − Vj / i j= MDi where POP(i) ≤ 1; ≤ 1; i = 1, C, (11) i = 1, C (12) 8: f (z) = Compute the fitness value of particles: ( /1 + SEP(z)) + ( /1 (13) + OUT (z)) 9: 10: 11: 12: velocityij = c1 ∗ velocityij + c2 ∗ rand(−1, 1) ∗ (zpBest,j − zij ) + c3 ∗ rand(−1, 1) ∗ (zgBest,j zij = zij + velocityij , c1 + c2 + c3 = 13: 14: context variable so-called Context-FGWC2 to get the crisp center V(1) (Machine 2) and V(2) (Machine 3) If the difference between the initial and crisp centers is smaller than a threshold (Eps) or the maximal number of iterations (MaxStep) is reached then we stop the Context-FGWC2 procedure and take the crisp center and membership degree, i.e U(1) , V(1) (Machine 2) and U(2) , V(2) (Machine 3) as the final results Otherwise, we assign V(0) = V(1) in Machine and V(0) = V(2) in Machine and start a new iteration in Context-FGWC2 until the stopping conditions hold Once the upper and lower centers and membership degrees are calculated, we use a defuzzification method so-called the Partition Coefficient and Exponential Separation (PCAES) [40] validity index to obtain the final center and membership degree as below V (∗) = V (1) if PCAES(V (1) ) ≥ PCAES(V (2) ) V (2) otherwise (20) C PCAES[j] j=1 where ⎛ N ukj PCAES[j] = k=1 ⎜ uM − exp ⎝ −min{ Vj − Vi i= / j ˇT } ⎞ ⎟ ⎠, (22) N u2ki uM = 1≤i≤C (23) k=1 ˇT = C l=1 Vl − V C (24) V = (V , V , , V r ) where V i (i = 1, r) is calculated as, This index measures the potential, whether the identified cluster has an ability to be a good cluster or not It was compared with other indexes such as Partition Entropy (PE), Partition Coefficient (PC), Fuzzy Hypervolume (FHV), Xie & Beni, Pal & Bezdek, Modification PC (MPC), Zahid et al., and showed the impressive results, even in a noisy environment The definition of PCAES is given below PCAES(C) = Calculate pBest and gBest as in the traditional PSO algorithm [18] End For For each particle Update new velocity and position: − zij ), (14) (15) (16) End For Until MaxStep PSO (21) Vi = C V l=1 li C (25) PCAES[j] is used to measure the compactness and separation for cluster j (j = 1, C) They are summed up to calculate PCAES(C) ∈ (− C, C) The large PCAES(C) value means that each of these C clusters is compact and separated from other clusters It is a criterion to choose the suitable clustering’s output Depending on which center is opted, the related membership degree is used as final membership U(*) Now, we describe the Context-FGWC2 procedure Remembering in Section “The interval context” that an interval context was defined as f = [f1 ,f2 ] so that we could apply fi (i = 1,2) in each machine Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Table The pseudo-code of Context-FGWC2 procedure - Initial center V(0) , the pattern set X, an interval fuzzifier [m1 ,m2 ], - The number of elements (clusters) – N(C), the dimension of dataset r, - Geographic parameters ˛, ˇ, a and b, precision ε, MaxStep iteration - Final center V(3) Input Output Context-FGWC2 V(3) ← V(0) Repeat V(0) ← V(3) 1: 2: 3: Compute U(x) = U(x), U(x) by (26)–(29) 4: V(A) ← V(0) For l = 1, r: Sort X following by lin ascending order Find index k0 satisfying (30) Otherwise, k0 ← N − Calculate U(1)(l) , V(1) by (31)–(32) If V(1) = V(A) 5: 6: 7: 8: 9: 10: For s = l + 1, r: Ukj (1)(s) ← Ukj (j = 1, C, k = 1, N) Go to Step 16 Else V(A) ← V(1) End If End For VR ← V(1) Calculate U(1) by (33) Repeat from Step to 17 to calculate VL , U(2) Perform Type-Reduction by (36) Determine the population of each cluster by (37) Update U(C) (x) by geo-characteristics in (2), (3) and (38)–(40) Perform Type-Reduction and compute center V(2) by (41) and (42) to get UGT (x) V(B) ← V(2) Repeat from Step to 18 to calculate VR , VL from V(B) and UGT (x) Perform defuzzification to calculate V(3) by (43) 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: V (3) − V (0) Until ≤ ε or MaxStep is reached Specifically, f1 (f2 ) was used in the Context-FGWC2 procedure of Machine (3) Because of using different context values and initial centers in those machines, the upper and lower centers and membership degrees totally reflect the basic principle of IT2FS The basic idea of the Context-FGWC2 procedure in Machine is using an interval of primary membership consisting of the lower and upper ones calculated from the initial center and updating the interval by geo-characteristics and context value f1 The pseudo-code of Context-FGWC2 is shown in Table In Step of the Context-FGWC2, the intervals of primary membership consisting of the upper and lower memberships are calculated by Eqs (26)–(29) Notice that in (26)–(27), the sum of membership degrees in all clusters is equal to f1k where f1k is a context value of the kth point in the pattern set Analogously, the values of the upper and lower memberships are depended by this context value as shown in (28)–(29) U(x) = U(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ukj = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ ⎨ C Ukj ∈ (0, 1)|k = 1, N; j = 1, C; ⎩ Ukj = f1k j=1 ⎧ ⎨ C Ukj ∈ (0, 1)|k = 1, N; j = 1, C; ⎩ Ukj = f1k j=1 f1k C i=1 Xk − Vj (0) 2/m1 −1 , if Xk − Vi (0) C i=1 Xk − Vj (0) Xk − Vi (0) 2/m2 −1 , (26) ⎭ Ukj = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (27) ⎭ Xk − Vj (0) 2/m1 −1 , if Xk − Vi (0) i=1 C Xk − Vj (0) f1k C i=1 f1k 2/m2 −1 , < 1/C Xk − Vj (0) Xk − Vi (0) (29) otherwise Xk − Vi (0) i=1 After we have the interval of primary membership, the maximum (minimum) center VR (VL ) and the related membership matrix U(1) (U(2) ) are calculated by the same steps from Step to 17 Specifically, in Step index k0 in the range [1,N − 1] satisfying Eq (30) will be selected as a pivot to calculate U(1)(l) in Eq (31) Xk0 l ≤ C v (A) j=1 jl (30) C ≤ X(k0 +1)l Ukj (1)(l) = (1) Vji Ukj if k ≤ k0 Ukj otherwise , (j = 1, C, k = 1, N) (31) = [m1 +m2 /2] N Xki (Ukj (1)(l) ) k=1 , [m +m /2] N (Ukj (1)(l) ) k=1 (j = 1, C, i = 1, r) (32) Next, in Step 10 we check whether V(1) = V(A) or not If this condition holds, we conclude that the maximum center VR = V(1) and the related membership matrix U(1) is found in Eq (33) ≥ 1/C Xk − Vj (0) Xk − Vi (0) (28) otherwise f1k C Using the average operator of fuzzifier, center V(1) is calculated below ⎫ ⎬ f1k C i=1 f1k ⎫ ⎬ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ U (1) = r U (1)(l) l=1 r (33) Otherwise, we make another loop with the next feature l in the pattern set By the similar process, in Step 18 we can compute the Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx (2) Ukj G = ˛ × Ukj + ˇ × × A C (2) wji × Uki (39) i=1 G (1) Ukj = ˛ × Ukj + ˇ × × A C (1) wji × Uki , (40) i=1 (i, j = 1, C, i = / j, k = 1, N) Notice that parameter A in Eqs (39) and (40) is a factor to scale the “sum” term and is calculated across all clusters, ensuring that the sum of the memberships for a given area k for all clusters is equal to the context value f1k (k = 1, N) Step 22 performs the typereduction for the modified membership degree and calculates new center V(2) by Eqs (41) and (42), respectively G Ukj GT = Vji (2) = Ukj + Ukj G , (j = 1, C, [m1 +m2 /2] N (Ukj GT ) Xki k=1 , N GT [m1 +m2 /2] (Ukj ) k=1 k = 1, N), (j = 1, C, (41) i = 1, r) (42) Now, we have modified membership degree UG and crisp center V(2) Since we work on IT2FS, V(2) should be an interval containing the minimum and maximum centers VL , VR This work is done through Step 23 and 24 In order to verify whether the outputted centers is the solution or not, Step 25 performs the defuzzification for the interval center as in Eq (43) and get crisp one V(3) This center is used to check the stopping condition described in Step 26 V (3) = Fig The mechanism of CFGWC2 minimum center VL and the related membership matrix U(2) where Eqs (31) and (33) are replaced with (34) and (35), respectively (2)(l) Ukj = U (2) = U (C) = Ukj if k ≤ k0 Ukj otherwise , (j = 1, C, k = 1, N) r U (2)(l) l=1 (34) (35) r U (1) + U (2) (36) From these related membership matrices, Step 19 obtains the membership degree of traditional fuzzy sets (a.k.a type-1) through Eq (36) This process is called the type-reduction and used to calculate the population of each cluster Step 20 calculates the population of each cluster by this rule: (C) (C) If Ukj > Uki and i = / j then Xk is assigned to cluster j, (37) (k = 1, N; i = 1, C) Based on the population, Step 21 determines the geographical weights of all areas by Eq (3), and the modification of membership degree following by geo-characteristics is performed through Eqs (2), (3) and (38)–(40) U G (x) = G(U (C) (x)) = Ukj G , Ukj G , (j = 1, C, k = 1, N) (38) VL VR if VL − V (0) ≤ VR − V (0) (43) otherwise In order to avoid unstoppable iteration, we limit the maximal number of iteration steps to MaxStep If the number of iteration steps exceeds this threshold, the Context-FGWC2 procedure will stop immediately Once the stopping condition holds, we receive the type-2 membership degree UG and the interval center [VL ,VR ] The crisp center V(3) and the distribution of pattern set after clustering can be extracted from them (UG ,V(3) ) are the output of Context-FGWC2, and the crisp center V(3) is denoted in Fig as V(1) (Machine 2) and V(2) (Machine 3) The works of Context-FGWC2 in Machine is analogous to those in Machine except the maximal number of iteration steps in Machine is equal to half of that in Machine (∼MaxStep/2) The reason for this alteration lies in the synchronization process Specifically, the results in Machine and are transferred to Machine after completion so that if a machine takes too much time to generate the outputs, it will cause large delayed time of the overall system Because the initial center of Machine is somehow better than that of Machine 2, the convergence may be faster and is not affected by the number of iteration steps In practical, the number of machines can be reduced, for instance the works of the Machine can be assigned to one of two left machines Because it takes much time to transfer data between machines, it is better if we can decrease the waiting time If so, the number of transferred steps between machines is reduced by half and the overall processing time is reduced remarkably The advantages of CFGWC2 are fourth-fold: Firstly, it is capable to handle the bad initialization and immature convergence by the PSO procedure; secondly, the clustering results focus on the users’ purposes by the interval context; thirdly, the computing speed of CFGWC2 is ameliorated through the interval context and the parallel computing mechanism; fourthly, the most important advantage of CFGWC2 is the high clustering quality in comparison with some relevant methods since this algorithm was deployed on Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ASOC-2296; No of Pages 19 ARTICLE IN PRESS L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Fig The two-dimensional distribution of UNO dataset IT2FS, which is more general and able to handle the existing limitations of the traditional fuzzy sets The disadvantage of CFGWC2 could be the computational costs and its complex activities Nevertheless, by employing some additional techniques we hope that the disadvantages could be ameliorated, and CFGWC2 achieves good clustering results Results Experimental environment This section describes the experimental environment used in next ones • Experimental tools: We have implemented the proposed algorithm (CFGWC2) in addition to these algorithms: NE [13], FGWC [24] and CFGWC [35] in MPI/C programming language and executed them on a Linux Cluster 1350 with eight computing nodes of 51.2 GFlops Each node contains two Intel Xeon dual core 3.2 GHz, GB Ram The experimental results are taken as the average values after 10 runs • Cluster validity: We use PCAES validity function described in Eqs (21)–(25) • Dataset: We use two kinds of datasets below - A real dataset of socio-economic demographic variables from United Nation Organization (UNO) [39] containing the statistic about population of 230 countries over ten years (2001–2010) Missing data were processed by Binning method [16] The twodimensional distribution is illustrated in Fig - A benchmark demographic dataset from The University of Edinburgh, Scotland (Fig 3) including expression levels of 2880 genes taken in 11 different areas [7] This dataset was used in many different research papers on gene expression by geographical factors such as in [4,5] • Objective: We compare the clustering quality of CFGWC2 with those of other algorithms through PCAES index Additionally, the Fig The two-dimensional distribution of Colon Cancer dataset Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Table PCAES values of all algorithms in Case on UNO dataset C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 1091.30832 3508.71041 1026.1004 851.56196 734.85210 11.49441 14.20249 9.66077 13.83029 23.45840 106.87815 102.97090 101.00239 98.86012 105.61367 106.87815 103.08807 101.05883 98.89076 105.11415 730.86493 1764.55205 1882.45315 828.00298 713.06259 15.80779 15.48401 9.60082 20.09243 13.36007 107.95304 104.51216 102.01264 98.70007 106.82538 107.95304 104.62430 102.07279 98.73446 95.32594 C m = 2.5 m = 3.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 435.14908 699.52639 758.04253 729.73602 660.41492 15.35085 17.05059 12.13725 13.80425 21.53153 110.80574 112.36477 111.70188 109.59175 107.14039 110.80576 112.46454 111.77472 109.64291 107.19830 222.59648 448.65676 530.12028 544.21607 534.99351 14.84918 18.15664 15.16747 17.33470 18.78905 111.54395 121.39454 123.22859 122.96865 122.06920 111.54397 121.45259 123.30832 123.03807 123.31178 Fig Average PCAES of algorithms on UNO dataset by fuzzifiers evaluation about the computational times of these algorithms is also mentioned Evaluation by various case studies In this section, we evaluate the proposed algorithm in comparison with the relevant methods by various case studies about the parameters of algorithms Main findings are found below Case In this case, some parameters of these algorithms are set up as below - The default geo-characteristics are: a = b = 1, ˛ = 0.7, ˇ = 0.3 These values determine the geo-modification process stated in Eqs (1)–(3) Our previous work [35] suggested using value ˛ ≥ 0.6 in order to increase the clustering quality - We use the default context values in [35] for CFGWC algorithm below where fi = ⎧ ⎪ ⎨ f = (f1 , f2 , , fN ), if k = ⎪ ⎩ rand(0, 1) otherwise k , k = imod4, i = 1, N (44) - In CFGWC2, m2 = × m1 = × m where m is the fuzzifier of NE, FGWC and CFGWC The interval context f = f , f where f1 = f and f2 = A broad interval of fuzzifiers and contexts will create more distinct results than a narrow one - In PSO, MaxStep PSO = 100 and population size is 500 Other parameters are (c1 , c2 , c3 ) = (0.2, 0.3, 0.5) and ( , ) = (1, 1) As suggested by Thien et al [38], these values will make the convergence to the optimum faster - Threshold ε and MaxStep of all algorithms are 10−3 and 500, respectively Table describes the PCAES values of all algorithms on UNO dataset The experiments are performed following by different values of the number of clusters and fuzzifiers Results show that PCAES values of CFGWC2 are the largest among all This means that the clustering quality of CFGWC2 is better than those of other algorithms In order to comprehend the experimental results, we illustrate the PCAES values of all algorithms through various cases of fuzzifiers in Fig From this figure, we recognize that PCAES values of CFGWC2 are larger than those of other algorithms For example, PCAES of CFGWC2 in Fig is 13 times greater than that of FGWC when m = 1.5 These numbers in cases of NE and CFGWC are 14 and 99 times, respectively Similarly, when m = 3.0, PCAES of CFGWC2 is still larger than those of other algorithms, i.e 3.79 (FGWC), 3.78 (NE) and 27 times (CFGWC) These evidences confirm that the clustering quality of CFGWC2 is the best among Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Table The computational time of all algorithms in Case on UNO dataset (s) C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 7.68 14.55 12.94 11.14 20.94 0.04 0.03 0.07 0.07 0.07 0.04 0.09 0.08 0.16 0.24 0.03 0.11 0.12 0.12 0.19 10.165 14.31 12.86 17.49 24.56 0.04 0.04 0.08 0.07 0.11 0.04 0.10 0.11 0.17 0.30 0.04 0.13 0.14 0.14 0.22 C m = 2.5 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 5.23 14.98 15.96 17.57 24.82 0.03 0.04 0.09 0.11 0.17 0.04 0.08 0.17 0.19 0.31 0.03 0.15 0.21 0.19 0.36 10.06 15.40 18.06 22.02 24.87 0.04 0.06 0.11 0.27 0.23 0.04 0.09 0.19 0.23 0.36 0.04 0.12 0.17 0.18 0.30 m = 3.0 all Nonetheless, PCAES values of CFGWC2 tend to decrease when the fuzzifier increases For instance, PCAES values of CFGWC2 from m = 1.5 to m = 3.0 are 1442, 1183, 656 and 456, respectively The average reducing ratio per half of a fuzzifier is 31% This means that each time the value of fuzzifier is increased by 0.5, PCAES value of CFGWC2 is reduced by 31 percents on average On the other hands, the average PCAES values of other algorithms seem to be stable through different values of fuzzifier, i.e 109 (FGWC), 108 (NE) and 15 (CFGWC) By rough calculation, we can easy find the value of fuzzifier that makes PCAES value of CFGWC2 is smaller than other algorithms, i.e m ≥ 5.0 This fact tells us the truth that CFGWC2 should be used when the fuzzifier is small As mentioned by Bezdek et al [3] when designing FCM algorithm, the authors stated that the fuzzifier should be from 1.5 to 2.5, ideally m = 2.0, for the sake of optimal centers found by the algorithm Thus, we may see that some cases such as m ≥ 5.0 will never happen in practical applications However, this finding may be useful for us to choose the appropriate value of parameters Is there any change of the order of algorithms in terms of PCAES values by different values of number of clusters? Following by Table 3, the answer is absolutely no For a given number of clusters, PCAES value of CFGWC2 is always larger than those of algorithms Indeed, this shows the stability of the proposed algorithm The computational time of all algorithms for exporting the results in Table is described in Table Clearly, the computational time of CFGWC2 is longer than those of other algorithms When m = 3.0, the average computational time of CFGWC2, FGWC, NE and CFGWC are 18.1, 0.182, 0.162 and 0.142 s, respectively Similar results are obtained in m = 2.0 and m = 2.5 As we may see in the pseudo-code of Context-FGWC2, it requires huge computation to process the interval membership matrix By using some additional techniques to speed up this algorithm, the computational time of CFGWC2 is reduced remarkably The maximal (minimal) computational time of CFGWC2 in Table is 24.87 (5.23) s With the increasing of computing powers nowadays, the computational cost in this case is acceptable Table also gives us the average increment levels of the computational time of algorithms per fuzzifier Each time the fuzzifier is increased by one unit, the computational time of CFGWC2 is increased by 16.8 percents The percent values of FGWC, CFGWC and NE are 29.5%, 57% and 64.9%, respectively When the fuzzifier is large enough, these times could be approximate to the others Now, we evaluate the proposed algorithm on a larger dataset than UNO In Fig 5, we measure the average PCAES values of all algorithms on Colon Cancer dataset following by fuzzifiers The results show that PCAES values of CFGWC2 are larger than those of other algorithms For example, when m = 1.5, the average PCAES value of CFGWC2 is 1.13 times larger than that of CFGWC These numbers in cases of FGWC and NE are 2.2 and 2.19 times, respectively Similarly, when m = 3.0, the average PCAES of CFGWC2 is 1.32 times, 1.15 times and 1.16 times larger than those of CFGWC, FGWC and NE, respectively These evidences confirm that the clustering quality of CFGWC2 is the best among all even on a large dataset such as Colon Cancer Nonetheless, PCAES values of CFGWC2 and other algorithms tend to decrease when the fuzzifier increases The values of CFGWC2 from m = 1.5 to m = 3.0 are 48.77, 34.18, 26.95 and 22.94, respectively This result is similar to that on the UNO dataset and shows that we should choose the small value of fuzzifier in this case in order to obtain good clustering quality of CFGWC2 Even when PCAES values of CFGWC2 reduce, they are still better than those of other algorithms The average PCAES value of CFGWC2 is approximately 1.4 times larger than those of other algorithms through various cases of fuzzifiers This means that when the fuzzifier increases, PCAES values of both CFGWC2 and other algorithms reduce, but the values of CFGWC2 are still larger than those of other algorithms However, small PCAES values of CFGWC2 in cases of large fuzzifier are not a good choice for us, and we should keep the fuzzifier is as small as possible In Fig 6, we verify whether or not PCAES values of CFGWC2 are larger than those of other algorithms by the number of clusters This figure clearly points out that the line of PCAES values of CFGWC2 is higher than those of other algorithms The started point of all lines (C = 2) shows that PCAES values of algorithms are approximate to the others, i.e 7.87 (CFGWC2), 8.67 (CFGWC), 7.182 (FGWC) and 7.184 (NE) However, the differences between those lines are getting obvious when the number of clusters increases For example, when C = 3, PCAES values of CFGWC2, CFGWC, FGWC and NE are 23.4, 19.3, 16.67 and 16.62, respectively When C = 6, the difference between CFGWC2 and other algorithms is maximal since the amplitudes of those lines expand PCAES values of those algorithms in this case of clusters are 56.2, 47.5, 33.8 and 33.2, respectively Thus, three remarks are extracted from this figure: (i) the clustering quality of CFGWC2 is the best even when all algorithms are tested following by the number of clusters; (ii) The higher the number of clusters is, the larger PCAES value of CFGWC2 is; (iii) The value of fuzzifier should be inversely proportional to that of the number of clusters for the sake of high PCAES values of CFGWC2 as shown in Figs and In Fig 7, we verify the changes of PCAES values of CFGWC2 by fuzzifiers on various datasets Clearly, PCAES values on a large dataset (Colon Cancer) are much smaller than those on small dataset (UNO) For example, the average PCAES values of CFGWC2 on UNO and Colon Cancer are 1442 and 48.77, respectively when m = 1.5 Similar results can be seen when m = 3.0 with PCAES values on UNO and Colon Cancer being 456 and 22.94, respectively Thus, two remarks are found from this test: Firstly, the sizes of inputted datasets should be small or medium for the high PCAES values of CFGWC2; secondly, the changes of PCAES values through various fuzzifiers on a large dataset are smaller than those on a small one Running on a large dataset such as Colon Cancer results in high computational time of CFGWC2 as shown in Fig This figure compares the average computational time of CFGWC2 on UNO and Colon Cancer datasets by fuzzifiers The average processing time of CFGWC2 per fuzzifier on Colon Cancer is 418 s whilst that processing time on UNO is 15.7 s From this result, we should consider the first remark about small or medium inputted datasets when running CFGWC2 algorithm The major remark in this case is the confirmation of the best clustering quality of CFGWC2 among all Case In Case 2, we make some changes of the parameters of all algorithms Specifically, geo-characteristics are ˛ = 0.4 and ˇ = 0.6 Other parameters are kept intact as in Case The aim is to verify Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ASOC-2296; No of Pages 19 10 ARTICLE IN PRESS L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Fig Average PCAES of algorithms on Colon Cancer dataset by fuzzifiers Fig Average PCAES of algorithms on Colon Cancer dataset by number of clusters Fig Changes of PCAES values of CFGWC2 by fuzzifiers on various datasets Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 11 Fig Average computational time of CFGWC2 on UNO and Colon Cancer datasets whether the clustering quality of the proposed algorithm is better than that of others or not when ˛ value (geographic parameter) is smaller than that of Case The results in Table show that PCAES values of CFGWC2 are still the largest among all of other algorithms For example, when m = 1.5, the average PCAES value of CFGWC2 is 959 It is 9.42, 9.44 and 28.4 times larger than those of NE, FGWC and CFGWC, respectively Similar results are found with three left cases of fuzzifier in which the PCAES values of CFGWC2 are still larger than those of other algorithms Thus, the change of geographic parameters does not affect the outcome results of algorithms Now, we investigate the impact of reducing the value of ˛ parameter to PCAES values of all algorithms Firstly, the average PCAES values of CFGWC2 per the number of clusters not reduce when the fuzzifier increases For example, these values in cases from m = 1.5 to m = 3.0 are 959, 877, 1144 and 696, respectively In Table 3, we got a remark that CFGWC2 should be used when the fuzzifier is small Nonetheless, it does not hold in this case since the reduction of ˛ value will increase the change of the membership degree of an area following by other ones’ as shown in Eq (1) As a result, PCAES value does not depend on the fuzzifier This fact shows that the changes of geographic parameters can help us reduce the dependence of CFGWC2 on the fuzzifier Secondly, the average PCAES values of CFGWC2 in this case are smaller than those in the previous one when m ≤ 2.0 and are larger than those in the previous one for the rests For example, PCAES values of CFGWC2 in Case and Case when m = 1.5 are 1442 and 959, respectively Nonetheless, these values in case of m = 3.0 are 456 and 696, respectively This means that reducing ˛ value will decrease the clustering quality of CFGWC2 Nevertheless, the reducing ratio of PCAES values is not as large as that of the previous case when the fuzzifier increases Each time the fuzzifier is increased by 0.5, PCAES values of CFGWC2 in Case and Case are reduced by 31% and 5.76%, respectively This explains why PCAES values of CFGWC2 in Case are larger than those in Case when m > 2.0 Thus, we should set the value of fuzzifier m > 2.0 when ˛ value decreases or ˛ < 0.5 for the large PCAES values in CFGWC2 algorithm Finally, the difference of PCAES values between CFGWC2 and other algorithms in Case is smaller than that in Case The maximal difference in Case is recorded at m = 1.5 when the average PCAES value of CFGWC2 is 9.42, 9.44 and 28.4 times greater than NE, FGWC and CFGWC, respectively In Case 1, the maximal difference is also recorded at m = 1.5 when the average PCAES value of CFGWC2 is 14, 13 and 99 times larger than NE, FGWC and CFGWC, respectively The minimal difference in Case is (6.25, 6.26, 11.67) for the list above These numbers in Case are (3.78, 3.79, 27.05), respectively Thus, the reduction of ˛ value makes the difference of PCAES values between algorithms be small The computational time of algorithms on UNO dataset in this case are described in Table Similar to previous case, the Table PCAES values of all algorithms in Case on UNO dataset C C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 1063.54223 1159.25575 999.55488 883.39827 691.62333 20.3629 33.53309 31.28929 30.07119 53.32656 106.61419 102.97252 101.34948 99.71663 97.22073 106.61419 103.20172 101.45983 99.77627 97.75520 856.68444 1070.81389 974.06185 823.12417 664.32020 20.06651 36.77730 36.35071 52.03082 60.78891 107.24664 103.81761 101.77679 99.33932 96.34740 107.24664 104.03909 101.89110 99.40264 96.57951 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 617.29692 2612.09686 890.07623 813.10817 790.12919 20.00514 42.52125 49.62835 67.13987 78.78573 109.12593 108.10623 106.85262 104.97583 102.57698 109.12593 108.30283 106.98115 105.06321 102.95150 427.75450 974.07089 755.75772 691.34934 632.35675 19.79795 48.02871 62.70671 80.02289 87.51471 110.32253 112.83833 112.48978 111.26262 108.95160 110.32243 112.97839 112.62659 111.37412 109.66323 m = 2.5 m = 3.0 Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 12 Fig Average PCAES of algorithms in Case on Colon Cancer dataset by fuzzifiers Table Computational time of all algorithms in Case on UNO dataset (s) C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 9.00 13.30 8.94 9.75 16.12 0.01 0.02 0.03 0.06 0.09 0.02 0.07 0.07 0.06 0.19 0.03 0.09 0.12 0.09 0.12 4.37 15.17 7.69 11.28 25.16 0.02 0.02 0.04 0.05 0.05 0.03 0.06 0.07 0.14 0.11 0.04 0.29 0.13 0.15 0.15 C m = 2.5 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 5.63 11.50 9.78 11.02 25.42 0.03 0.04 0.05 0.09 0.12 0.03 0.08 0.08 0.10 0.12 0.04 0.17 0.09 0.12 0.13 9.48 14.57 13.17 22.22 25.56 0.03 0.03 0.06 0.18 0.23 0.03 0.07 0.09 0.10 0.12 0.04 0.10 0.09 0.11 0.16 m = 3.0 computational time of CGWC2 is larger than those of other algorithms Nonetheless, the average computational times of CFGWC2 through various fuzzifiers are smaller than those of Case From m = 1.5 to m = 3.0, these values in Case and Case are (13.45, 15.87, 15.71, 18.08) and (11.42, 12.73, 12.67, 17), respectively Therefore, reducing ˛ value makes CFGWC2 run faster In Fig 9, we verify the effectiveness of CFGWC2 on Colon Cancer dataset by comparing the average PCAES values of all algorithms following by fuzzifiers This figure clearly shows that PCAES values of CFGWC2 are larger than those of other algorithms The maximal difference of PCAES values between those algorithms is recorded at m = 2.0 when the average PCAES value of CFGWC2 is 4.87 times, 4.67 times and 4.93 times larger than those of CFGWC, FGWC and NE, respectively The minimal difference is at m = 3.0 when those equivalent values are 2.28, 2.11 and 2.19 times PCAES values of CFGWC, FGWC and NE are approximate to the others in this case with the domain of values belonging to the interval [22.18, 25.49] as shown in the figure Obviously, the clustering quality of CFGWC2 is still the best among all even though some changes of geographic parameters and datasets have been done In Fig 10, we study the changes of average PCAES values of CFGWC2 with different datasets and cases The aim of this test is to investigate the impact of geographic parameters and datasets to PCAES values of CFGWC2 Results show that PCAES values of CFGWC2 in this case are larger than those of Case of Colon Cancer dataset For example, when m = 1.5, the average PCAES values of CFGWC2 in Case and Case are 79.6 and 48.7, respectively In m = 2.0, the difference of PCAES between those cases are maximal with PCAES values being 119 (Case 2) and 34.1 (Case 1) This means that the change of geographic parameter, especially reducing the value of ␣, enhance PCAES values of the proposed algorithm Nevertheless, PCAES values of CFGWC2 on Colon Cancer dataset are much smaller than those on UNO When m = 2.5, the average PCAES value of CFGWC2 in Case of UNO dataset is 1144.5 These values in cases of Case and Case of Colon Cancer are 26.95 and 60.27, respectively Similar results are found with other cases of fuzzifiers Obviously, using small datasets obtains better PCAES values of CFGWC2 than large ones Is there any change of computational time of CFGWC2 with different cases and datasets? Fig 11 helps us answer this question by drawing three lines represented for the computational time of CFGWC2 in Case of Colon Cancer dataset (gray line), in Case of UNO (blue, dot line) and in Case of Colon Cancer dataset (green, double dot line) This figure states that using low values of geographic parameters (˛) in CFGWC2 reduces the computational time of this algorithm The proof for this consideration is that the line of “Case – Colon Cancer” is always lower than that of “Case – Colon Cancer” However, the “Case – Colon Cancer” line is much higher than that of “Case – UNO” Since the size of Colon Cancer dataset is 14 times larger than that of UNO, this increases the computational time of CFGWC2 as shown in the figure Even in this situation, the computational time of CFGWC2 is not much higher than those of other algorithms in this case because these times increase concurrently Thus, the computational time of CFGWC2 is acceptable in this situation In short, the changes of geographic parameters in this case not affect the order of algorithms in terms of clustering quality, and the clustering quality of CFGWC2 is proved to be the best among all Case In this case, we narrow the interval context and the interval fuzzifier Specifically, the interval fuzzifier of CFGWC2 is [m1 , m2 ] = [m, 1.5 × m] where m is the fuzzifier of NE, FGWC and CFGWC The interval context is f = f , f where f2 (f1 ) is the maximal (minimal) value between the function in Eq (44) and the standard Gaussian function in Eq (45) Other parameters are kept intact as in Case f = (f1 , f2 , , fN ), where fi = √ e−1/2i , (i = 1, N) 2˘ (45) Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 13 Fig 10 Changes of PCAES values of CFGWC2 in Case with different datasets & cases Fig 11 Changes of computational time of CFGWC2 in Case by datasets & cases Table PCAES values of all algorithms in Case on UNO dataset C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 139.95086 568.21771 448.02083 988.99686 6640.59369 6.04220 51.29265 225.29319 326.01488 259.32112 106.87815 102.97089 101.00240 98.86013 104.96678 106.87815 103.08807 101.05884 98.89074 105.43589 5258.34615 15,285.74240 292.73635 1098.36009 7153.92664 5.76239 52.38541 171.86304 134.54998 1286.04887 107.95304 104.51216 102.01262 98.70013 95.29829 107.95304 104.62430 102.07281 98.73448 95.32741 C m = 2.5 m = 3.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 1577.22397 365.63816 478.69445 15,865.95189 617.06103 8.16377 53.58911 353.06716 376.79995 165.71077 110.80570 112.36477 111.70188 109.59178 107.14640 110.80575 112.46454 111.77475 109.64291 107.19815 499.29655 1861.65345 2435.806574 4064.86640 15,167.8462 7.53170 63.74519 415.55560 285.29656 332.15287 111.54397 121.39453 123.22859 122.96855 121.56386 111.54390 121.45250 123.30832 123.03813 121.64574 Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 14 Table Computational time of all algorithms in Case on UNO dataset (s) C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 4.83 12.84 13.95 18.27 17.09 0.01 0.02 12.97 0.02 13.37 0.04 0.09 0.10 0.19 0.29 0.03 0.14 0.15 0.19 0.20 5.18 9.85 15.72 18.85 19.49 0.01 13.12 15.79 12.42 14.27 0.05 0.08 0.13 0.14 0.44 0.04 0.43 0.18 0.15 0.37 C m = 2.5 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 5.59 11.93 15.47 17.22 20.11 0.01 0.02 13.24 11.52 13.71 0.03 0.09 0.14 0.16 0.43 0.05 0.13 0.20 0.18 0.32 7.89 10.51 17.52 15.96 19.70 0.01 0.03 13.34 12.45 13.88 0.04 0.09 0.16 0.17 0.33 0.06 0.14 0.30 0.23 0.30 m = 3.0 The results of algorithms with the new configuration are illustrated in Tables and Table mentions PCAES values whilst Table shows the computational time of algorithms PCAES values of algorithms in Table point out that the clustering quality of CFGWC2 is the best among all With m = 1.5, the PCAES values of (CFGWC2, CFGWC, FGWC and NE) are (1757, 173, 102, 103), respectively Analogously, when m = 3.0, these values are (4805, 220, 120.13, 120.19), respectively This clearly shows that CFGWC2 still obtains the best clustering quality among all even though the interval context and the interval fuzzifier have been narrowed Some changes of PCAES values of algorithms in this case are herein highlighted Firstly, PCAES values of CFGWC2 are directly proportional to the fuzzifier For example, when m = 1.5, the average PCAES of CFGWC2 is 1757 When m increases to 2.5, PCAES of CFGWC2 is 3780 PCAES value is continued to increase to 4805 when m = 3.0 This result is opposite to that of Case when we receive a remark that the PCAES value of CFGWC2 tends to decrease when the fuzzifier increases Thus, we should set high value of fuzzifier with the configuration in this case in order to get good clustering quality of CFGWC2 Secondly, we compare the average PCAES values of CFGWC2 in Table with those in Table and get the remark that the values in Table are much higher than those in Table The pairs of PCAES values of CFGWC2 in (Table 3, Table 7) from m = 1.5 to m = 3.0 are (1442, 1757), (1183, 5817), (656, 3780) and (456, 4805), respectively Indeed, the impact of narrow context and fuzzifier really enhance the clustering quality of CFGWC2 as shown in the comparison above Thirdly, the difference of PCAES between CFGWC2 and other algorithms is smaller than that of Case Besides, this difference is stable through various fuzzifiers For example, the maximal difference between CFGWC2 and other algorithms is recorded at m = 3.0 when the average PCAES value of CFGWC2 is 21 times, 40 times and 39 times larger than those of CFGWC, FGWC and NE, respectively The minimal difference is recorded at m = 1.5 when the equivalent values are 10 times, 17 times and 17 times, respectively Comparing those results with ones in Case 1, we can recognize that the changes of narrow context and fuzzifier in CFGWC2 result in the stable difference between CFGWC2 and other algorithms Table shows the similar results with Table in Case when CFGWC2 runs longer than other algorithms The maximal and minimal computational times of CFGWC2 are 20.11 and 4.83 s, respectively Because these numbers are small, the computational cost of CFGWC2 can be acceptable In Fig 12, we illustrate the average PCAES of all algorithms on Colon Cancer dataset by fuzzifiers Intuitively, PCAES line of CFGWC2 is higher than those of other algorithms This clearly proves that the clustering quality of CFGWC2 is the best among all Besides, PCAES values of CFGWC2 not reduce when the fuzzifier increases This result is similar to that in Case 2, and is opposite to that in Case These evidences stress that the changes of geographic parameters, contexts and fuzzifiers can help CFGWC2 reduce the dependence on the fuzzifier Fig 13 shows the changes of PCAES values of CFGWC2 by different datasets and cases Similar to Case 2, the comparisons between the results in this case and those in Case on Colon Cancer and in Case on UNO dataset are highlighted The results point out that the average PCAES values of CFGWC2 in this case are much smaller than those in Case of UNO dataset The maximal and minimal PCAES values of CFGWC2 in Case are 5817 and 1757, respectively Those values in this case are 60.1 and 46.3, respectively Obviously, the difference of PCAES between two cases is quite large, even be larger than that in Case shown in Fig 10 Thus, the recommendation is that we should not use large datasets with the configuration of parameters in this case in order to avoid small PCAES values of CFGWC2 as such Nonetheless, PCAES values of CFGWC2 in this case and in Case on Colon Cancer are approximate to the others Fig 13 shows that the bars of these cases are nearly equal The maximal difference of PCAES between two cases is 32.7 Comparing with equivalent results in Fig 10, we may recognize that there is not much change of PCAES value if some modifications of fuzzifiers and contexts are performed like what were done in this case In Fig 14, we examine the changes of computational time of CFGWC2 by different datasets and cases Results show that the average computational time of CFGWC2 in this case is larger than those in Case on Colon Cancer This result is opposite to that of Case and tells us the fact that using new interval contexts and fuzzifiers makes CFGWC2 run slower than the algorithm without these configurations However, both the time of “Case – Colon Cancer” and “Case – Colon Cancer” are much slower than that of “Case – UNO”, which takes approximately 15 s on average to process a given value of fuzzifier Experiments with the changes of context and fuzzifier in Case re-confirm the superiority of CFGWC2 to other algorithms in terms of clustering quality Case The interval context in Case is near to zero value In this case, we perform the experiment with another interval context whose values are near to one f = (f1 , f2 , , fN ), ⎧ ⎨1 where fi = if k = ⎩ rand(0, 1) + 2k , (k = imod4, i = 1, N) otherwise (46) f = (f1 , f2 , , fN ), where fi = 1 +√ e−1/2i , 2˘ (i = 1, N) (47) The new interval context is defined as f = [f1 , f2 ] where f2 (f1 ) is the maximal (minimal) value between the function in Eq (46) and the modified Gaussian function in Eq (47) The interval fuzzifier of CFGWC2 is still [m1 , m2 ] = [m, 1.5 × m] Other parameters are kept intact as in Case Table describes PCAES values of all algorithms in Case on UNO dataset Results affirm the remark achieved in the previous cases in which PCAES values of CFGWC2 are much larger than those of other algorithms The average PCAES values of CFGWC2, CFGWC, FGWC and NE by the number of clusters and fuzzifiers are 1266, 116, 109 and 110, respectively Obviously, PCAES of CFGWC2 is 10.8 times, 11.57 times and 11.51 times higher than CFGWC, FGWC and NE, respectively Thus, the clustering quality of CFGWC2 is the best among all In order to investigate the impact of Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ASOC-2296; No of Pages 19 ARTICLE IN PRESS L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 15 Fig 12 Average PCAES of algorithms in Case on Colon Cancer dataset by fuzzifiers Fig 13 Changes of PCAES values of CFGWC2 in Case by different datasets & cases Fig 14 Changes of computational time of CFGWC2 in Case by datasets & cases Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ARTICLE IN PRESS ASOC-2296; No of Pages 19 L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 16 Table PCAES values of all algorithms in Case on UNO dataset C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 5045.66670 3875.38385 1558.83769 1304.13622 1122.92581 2.90326 357.41475 353.10979 351.67087 336.08991 106.87815 102.97089 101.00240 98.86010 104.96682 106.87815 103.08807 101.05883 98.89074 105.11414 1581.05213 3661.44151 1607.32553 1382.93286 1133.25453 3.00038 59.75581 84.23204 76.83167 244.14817 107.95304 104.51216 102.01262 98.70007 106.40734 107.95304 104.62430 102.07279 98.73447 118.10708 C m = 2.5 m = 3.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 832.81547 137.44910 132.35684 119.87765 109.09223 14.86731 66.55067 65.09628 62.66657 57.22595 110.80575 112.36477 111.70188 109.59177 107.14639 110.80574 112.46454 111.77476 109.64291 107.09815 489.21252 545.80669 364.85714 95.71117 227.36512 2.61660 46.76435 52.27548 48.18992 47.98189 111.54398 121.39451 123.22859 122.96870 123.39548 111.54395 121.45261 123.30831 123.03806 122.64568 the new interval context to PCAES values of CFGWC2, we calculate the average PCAES values from m = 1.5 to m = 3.0 such as (2581, 1873, 266, 344), respectively These results show two remarks: (i) Opposite to the remark in Case 1, PCAES values of CFGWC2 not reduce when the fuzzifier increases; (ii) PCAES values of CFGWC2 are large when the fuzzifier is small, i.e m ≤ 2.0 Otherwise, PCAES values are small The second remark is similar to that in Case Comparing the PCAES values of CFGWC2 in Table with those in Table 3, we recognize that when the fuzzifier is small (m ≤ 2.0), the values in Table are larger than those in Table Nevertheless, the results are reversed for the left cases of fuzzifier This result reflects the large distinction between PCAES values when m ≤ 2.0 and those when m ≤ 2.0 in this case Thus, a remark is extracted through this observation is that we should choose the fuzzifier m ≤ 2.0 with the configuration in this case in order to obtain high value of PCAES in CFGWC2 The maximal difference of PCAES between CFGWC2 and other algorithms is found at m = 2.0 when the average PCAES values of CFGWC2 is 20 times, 18 times and 17 times larger than those of CFGWC, FGWC and NE, respectively This difference is small in comparison with those in Table Indeed, using the new interval context results in the small difference of PCAES values between CFGWC2 and other algorithms We also measure the computational times of algorithms and describe them in Table 10 This table points out that the computational time of CFGWC2 is longer than those in Table The maximal and minimal computational times of CFGWC2 are 27.08 (m = 2.5) and 4.49 (m = 2.0), respectively The maximal value is larger than those of previous cases However, the minimal one is Table 10 Computational time of all algorithms in Case on UNO dataset (s) C m = 1.5 m = 2.0 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 12.16 14.56 18.82 23.82 24.53 0.02 0.06 0.08 0.17 0.24 0.03 0.10 0.12 0.14 0.20 0.03 0.10 0.14 0.15 0.24 4.49 14.35 16.48 22.46 24.86 0.03 0.09 0.18 0.22 0.32 0.04 0.10 0.23 0.16 0.35 0.03 0.19 0.14 0.12 0.21 C m = 2.5 CFGWC2 CFGWC FGWC NE CFGWC2 CFGWC FGWC NE 5.91 15.27 18.91 17.02 27.08 0.05 0.09 0.38 0.40 0.24 0.04 0.20 0.37 0.33 0.40 0.04 0.31 0.15 0.14 0.45 5.37 15.15 18.66 25.06 24.66 0.04 0.11 0.18 0.25 0.39 0.05 0.09 0.17 0.34 0.33 0.04 0.12 0.16 0.16 0.28 m = 3.0 the smallest among all Thus, the remark above about choosing m ≤ 2.0 is re-confirmed In Fig 15, we measure the average PCAES values of all algorithms on Colon Cancer dataset by fuzzifiers The results show that the average PCAES value of CFGWC2 is larger than those of other algorithms For example, PCAES values of CFGWC2, CFGWC, FGWC and NE are 55.51, 21.96, 23.08 and 21.90, respectively when m = 1.5 However, PCAES values of not only CFGWC2 but also other algorithms tend to decrease when the fuzzifier increases Thus, the differences of PCAES values between those algorithms are getting smaller When the fuzzifier is small enough, PCAES values of all algorithms are quite small In the other words, the clustering qualities of all algorithms are inversely proportional to the fuzzifier Thus, an important remark of this case is that we should not choose large values of fuzzifier in order to keep good clustering quality of CFGWC2 In Fig 16, we investigate the impact of parameters to PCAES values of CFGWC2 Obviously, using narrowed interval context and fuzzifier whose values are near to one as in this case not improve PCAES values of CFGWC2 significantly From m = 1.5 to m = 3.0, the average PCAES values of “Case – Colon Cancer” bar are not always larger than those of “Case – Colon Cancer” For example, when m=1.5, PCAES values of these bars are 55.51 and 48.77, respectively When m = 2.5, these values are 26.71 and 26.95, respectively We also draw another bar of “Case – Colon Cancer” to clearly recognize the impact of parameters Fig 16 points out that the average PCAES values of “Case – Colon Cancer” are not only better than those of “Case – Colon Cancer” but also better than those of “Case – Colon Cancer” This means that the impact of parameters in this case to PCAES values of CFGWC2 is not equal to that in Case The impact of datasets to PCAES values of CFGWC2 is illustrated in Fig 17 PCAES values of CFGWC2 in “Case – Colon Cancer” are much smaller than those in “Case – UNO” Thus, we also get the similar remark with that of previous cases In Fig 18, we compare the average computational time of CFGWC2 through various datasets and cases Through this figure, we recognize that CFGWC2 in “Case – Colon Cancer” runs slower than those in “Case – Colon Cancer” and in “Case – UNO” When m < 2.7, it is slower than that in “Case – Colon Cancer” Thus, the fuzzifier should be set small if the configuration of parameters in this case is used for CFGWC2 Summary of the findings In this section, we sum up the main findings in Section “Evaluation by various case studies” as follows: Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ASOC-2296; No of Pages 19 ARTICLE IN PRESS L.H Son / Applied Soft Computing xxx (2014) xxx–xxx 17 Fig 15 Average PCAES of algorithms in Case on Colon Cancer dataset by fuzzifiers Fig 16 Impact of parameters to PCAES values of CFGWC2 in Case Fig 17 Impact of datasets to PCAES values of CFGWC2 in Case Please cite this article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025 G Model ASOC-2296; No of Pages 19 18 ARTICLE IN PRESS L.H Son / Applied Soft Computing xxx (2014) xxx–xxx Fig 18 Changes of computational time of CFGWC2 in Case by datasets & cases • The clustering quality of CFGWC2 is the best among all even on a large dataset such as Colon Cancer • CFGWC2 is stable through various numbers of clusters and fuzzifiers • PCAES values of CFGWC2 are directly proportional to the number of clusters • In order to achieve the best clustering quality in CFGWC2, some parameters should be set up as follows Geographic parameters: ˛ < 0.5, fuzzifier m > 2.0, and the interval context and interval fuzzifier are narrowed as in Case • The changes of PCAES values of CFGWC2 by fuzzifiers on a large dataset are smaller than those on a small one • The sizes of inputted datasets should be small or medium for the high PCAES values of CFGWC2 • The computational cost of CFGWC2 can be acceptable Conclusions In this paper, we concentrated on improving the clustering quality of the state-of-the-art clustering algorithm so-called FGWC for the GDA problem A novel interval type-2 fuzzy clustering algorithm namely CFGWC2 deployed in an extension of the traditional fuzzy sets namely Interval Type-2 Fuzzy Sets was presented It integrated some additional techniques to speed up the whole algorithm such as the interval context variable, Particle Swarm Optimization and the parallel computing The experimental results by various case studies on two benchmark datasets showed that CFGWC2 obtained better clustering quality than other relevant algorithms The experiments also suggested us which values of parameters should be chosen for the best quality of the proposed algorithm Further works will examine CFGWC2 for handling very large datasets, partly classified and time-series datasets Additionally, some applications of the proposed method in 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as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025... article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025... article in press as: L.H Son, Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization, Appl Soft Comput J (2014), http://dx.doi.org/10.1016/j.asoc.2014.04.025