Abstract
Fuzzy c-means is an efficient algorithm that is amply used for data clustering. Nonetheless, when using this algorithm, the designer faces two crucial choices: choosing the optimal number of clusters and initializing the cluster centers. The two choices have a direct impact on the clustering outcome. This paper presents an improved algorithm called automatic genetic fuzzy c-means that evolves the number of clusters and provides the initial centroids. The proposed algorithm uses a genetic algorithm with a new crossover operator, a new mutation operator, and modified tournament selection; further, it defines a new fitness function based on three cluster validity indices. Real data sets are used to demonstrate the effectiveness, in terms of quality, of the proposed algorithm.
1 Introduction
Clustering has been widely applied in various disciplines, including medical sciences [9], computer sciences [44], bioinformatics [20, 34, 48], bankruptcy forecasting [17], astronomy [3], and weather classification [35, 36]. Clustering can be divided into two different types: crisp and fuzzy. The former, which supposes the classes are clearly separated, is a traditional technique. It assigns each object to only one class [28]. By contrast, the latter does not make assumptions about the separation of the classes. In addition, instead of allocating the object to a unique class, fuzzy methods assign membership degrees to which the objects belong to classes [9]. Therefore, these models allow, generally, a better description of the real data where the borders between the classes are often inaccurately defined.
Fuzzy c-means (FCM) [7] is a dynamic method. Objects can change the degrees of membership during the process of class training. As this technique is iterative, the outcome is sensitive to initialization [5]. As a consequence, the improper selection of initial centroids will generally lead to undesirable clustering results [25]. A simulated annealing algorithm [47], a Tabu search algorithm [2, 39, 52], a genetic algorithm (GA) [4, 12, 23, 26, 37, 53], an ant colony [27, 30, 31, 49], and an artificial bee colony [32, 33] are examples of heuristic methods that have been used over the last two decades to overcome the problem of initialization. An exhaustive review of these algorithms can be found in Ref. [15].
Another handicap has been that FCM needs the number of clusters as input parameter. Viable methods to overcome this drawback can be found in Refs. [4, 11, 12, 23, 30, 38, 45, 46, 53]. In addition, other versions of FCM based on iterative or recursive techniques attempt remedying this encumbrance [13, 19]. However, these techniques present the problems of time complexity.
As clustering data sets can be viewed as an optimization problem [14], we propose a novel technique to cluster data based on a GA [24]. A relevant advantage of this algorithm is its ability to deal with local optima by maintaining, diversifying, and comparing several candidate solutions simultaneously. However, the GA challenge is to maintain the balance between exploitation and exploration.
We address the application of a hybrid GA, called automatic genetic FCM (AGFCM), based on gravitational mutation [43], differential crossover [41, 42, 51], modified tournament selection, and the FCM algorithm. We consider in AGFCM the balance between exploitation and exploration of candidate solutions using the new genetic operators mentioned previously.
The performance of AGFCM has been tested on three real data sets from the University of California at Irvine (UCI) repository [10], and the results have been compared using other techniques. The rest of this paper is organized as follows. Section 2 presents a brief description of the FCM algorithm. Section 3 describes our proposed algorithm to solve data clustering problems. Experimental results and comparison to other available methods are discussed in Section 4. Finally, conclusions and future work are highlighted in Section 5.
2 Fuzzy c-Means
Many clustering methods are introduced in the literature. These can be classified into two methods: hard and fuzzy. In hard clustering algorithms, which are based on classical set theory, the object belongs to one class. In fuzzy clustering algorithms, objects can belong to all classes with different degrees of membership. This is appropriate for real-world data where boundaries between clusters are not well defined. That is why fuzzy clustering presents the advantage of dealing with overlapping clusters.
Let X={x1, x2, ..., xn}⊂ℜp be a set of n objects with dimension p. Partitioning X in c clusters can be defined by a matrix U=[ui,j]∈ℜn×c, which satisfies the following three conditions:
where uij is the membership degree of xi for the jth cluster.
The FCM algorithm optimizes the Jm criterion defined by
where
V(v1, v2, …, vc)∈ℜc×p and vj is the jth prototype;
m (1<m<∞) is a parameter used to control the level of fuzziness in the resulting clusters;
|| || is a norm to measure the distance between the jth prototype and the ith data point.
Bezdek showed that FCM always converges to a minimum of Jm under the following conditions [7]:
The pseudo-code of the FCM algorithm is given in Algorithm 1.
FCM pseudo-code.
| Data: Vector of objects X: (x1, x2, ..... xn) |
| Result: Prototypes
|
| Choose: |
| – 1<c<n |
| – m>1 |
| – tmax maximum number of iterations |
| – ϵ tolerance threshold |
| – Norm for clustering criterion Jm |
| – Norm for calculating errors Et=||Vt−Vt−1|| |
| Initialization: |
| – Prototypes V0 |
| – t←0 |
| While (Et>ϵ and t<tmax) do |
| t←t+1 |
| Calculate Ut by using Eq. (5) |
| Calculate Vt by using Eq. (6) |
3 Proposed Method
In this section, we describe the AGFCM clustering algorithm. This algorithm uses a GA that utilizes new operators and a new fitness function to evolve the number of clusters and to provide the initial centroids. The results of the GA phase are then used as an input in the FCM algorithm. The pseudo-code of the AGFCM algorithm is given in Algorithm 2. The AGFCM clustering algorithm is introduced in what follows.
General description of AGFCM.
| Data: Data set |
| Result: Best individual: number of clusters; prototypes |
| Initialization |
| for each individual i in the population do |
| Choose ci∈{cmin, ..., cmax}; |
| In data set, Choose randomly ci objects |
| end |
| while not termination_condition do |
| Fitness evaluation; |
| Modified enthusiasm selection MES(); |
| Differential crossover; |
| Gravitational mutation; |
| end |
3.1 Chromosome Representation
To encode a chromosome, we use a real-valued representation. The chromosomes represent the coordinates of the cluster center. If we consider Pi as the ith candidate solution, Pi={vi1, vi2, …,
3.2 Population Initialization
In the AGFCM clustering algorithm, an initial population is randomly generated. For each chromosome, a number of classes, ci, is randomly chosen between cmin and cmax. ci points from the data set are randomly chosen to initialize the chromosome. In this paper, cmin is set to 2 and the value
3.3 Fitness Evaluation
The fitness function of a candidate solution indicates how relevant a solution is. In this paper, the fitness function is based on three well-known clustering validity measures:
Davies-Bouldin (DB) index: The DB index is a function of the ratio of the sum of within-cluster scatter to between-cluster separation [16]. This index is defined as
(7)where c is the number of clusters. Si is the scatter within the ith cluster and dij is the distance between cluster centers vi and vj. Si is defined as
(8)Xi is the ith cluster and dij is defined as
(9)Xie and Beni (XB) index: Xie and Beni [54] introduced a validity measure and defined it as
(10)where ukj is the membership of the jth point to the kth cluster.
In Eq. (10), the numerator measures the compactness of fuzzy partition. The denominator measures the separation between clusters.
Partition entropy (VPE) index: partition entropy is a function of U proposed by Bezdek [6]. VPE is formulated as
(11)where U=[ui,j] is the matrix of membership degrees.
The fitness function consists of summing the three indices outlined using a weighting coefficient for each of them. The function has the following form:
where fi is the index of validity expressed above.
In this study, we chose the weighting coefficients as
where i∈{1, 2, 3} and
3.4 Selection Method
The modified enthusiasm selection (MES) [1, 29] is used as a selection operator. The MES is based on the tournament selection method. It is a technique that gives another chance to worst individuals in a population to compete with the best individuals in the evolving process. With a view to increase the fitness of those individuals, an enthusiasm coefficient λ is multiplied (mathematical meaning) with the old fitness value. After the enthusiasm individual has been selected, the MES put it to its raw fitness. MES also guards in each iteration the best individual [29]. The pseudo code of the selection method is given in Algorithm 3. In Algorithm 3, TabS represents an array of n individuals’ indices in the current population. TabR is an array holding the indices of individuals in a random order. TabT represents an array of k−1 individual fitness, where k is the tournament size. TabW, an array of individual indices, is the result of the selection operation.
3.5 Crossover Operator
A crossover operator is a probabilistic process to combine selected parents to create new offsprings. The crossover operator implemented in this study is a two-points crossover; it is a two-parents-two-offsprings schema following the steps below:
Firstly, two individuals P1 and P2 are randomly selected, and two crossing points are chosen.
The crossover occurs between P1 and P2 using simulated binary crossover and generates two intermediate children, C1 and C2. Note that crossover points are restricted to fall on the same location within each cluster description.
Differential crossover is used. It combines two strategies into one including their entire advantages [21, 42, 50]. The first strategy uses the values of the objective function to determine a “good” direction. The second strategy uses the best individual. Notice that the introduction of information such as the “good” direction and the best individual reduces the search space exploration capabilities. The new children are generated by the differential crossover according to
Algorithm 3:MES().
Data: Array: (TabR(), TabS() ) Result: Array: (TabW()) Initialization; k←c; l←0; for i←0 to k do Shuffle TabR(); j←0 while j<n do C1←TabR(j); for m←1 to k do C2←TabR(j+m); if f(C1)<f(C2) then C1←C2 end if m>1 then aux←m; for m←1 to k do TabT(m)←f(Ij+m); f(Ij+m)=λf((Ij+m); end m←aux; end end for m←1 to k do f(Ij+m)←TabT(m); end j←j+k+1; TabW(l)←C1; TabW(l+1)←TabS(l); l←l+2; end end (14)and
(15)λ is used to enhance the crossover when incorporating the current best vector xbest. F1 and F2, which control the amplification of the differential variation of the offsprings C1 and C2, are real and constant factors. In this paper, we set λ=F1=F2=0.9.
3.6 Mutation Operator
The mutation operator diversifies the population and avoids the creation of a set of homogeneous population elements. It should change the solution adequately to leave the attraction basin of the local optimum, but it should also avoid changing the solution too much and destroying already promising structures.
A novel mutation operator named gravitational mutation is used in this paper. The proposed operator is based on the gravitational search algorithm (GSA). GSA, a population-based search algorithm, is based on the law of gravity and interaction between masses [43]. To mutate an individual Pi, Eq. (16) is used:
where i ∈ {1, .... N}, N is the population size, and si is the next velocity of Pi computed as
randi is a random number in [0,1]. ai is the acceleration of Pi computed as
Here, Fi is the total force acting on Pi, and it is calculated as follows:
where randj is a random number in [0,1], G is the gravitational constant, Rij is the Euclidian distance between Pi and Pj, ϵ is a small constant to avoid division by zero, and Mi is the mass of Pi calculated as follows:
where fiti is the fitness value of Pi and worst is defined as follows (for a maximization problem):
and j ∈ {1, …, N}.
4 Experimental Results
To evaluate the relevance of AGFCM, experiments were conducted on three real-world data sets from the UCI Machine Learning Repository [10]: the Iris data set, Glass data set, Wine data set, and Breast Cancer Wisconsin data set.
The Iris data set [22]: This may be one of the most used data sets in clustering problems. It consists of three classes that represent three species of iris plants: Iris setosa, Iris virginica, and Iris versicolor. Fifty observations belong to each class. Each observation consists of four characteristics: length and width of the sepal and petal of the flower [10].
Glass data set: It consists of six classes that represent six different types of glass – building windows float processed, building windows non-float processed, vehicle windows float processed, containers, tableware, and headlamps. Each observation consists of nine numeric attributes: refractive index, sodium, magnesium, aluminum, silicon, potassium, calcium, barium, and iron [10].
Breast Cancer Wisconsin: It consists of two classes that represent benign (239 objects) or malignant (444 objects) tumors. Each observation consists of nine features: clump thickness, uniformity of cell size, uniformity of cell shape, marginal adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli, and mitoses [10].
The Wine dataset is the result of a chemical analysis of wines grown in three different cultivars in the same region in Italy [10]. There were 178 samples: 59 in the first class, 71 in the second class, and 48 observations belong to the third class. Each observation consists of 13 numeric features: alcohol, malic acid, ash, alkalinity of ash, magnesium, total phenols, flavanoids, non-flavanoid phenols, proanthocyanins, color intensity, hue, OD280/OD315 of diluted wines, and proline [10].
The following criteria were used to compare the clustering results:
The number of classes found.
Inter-cluster distance: the distance between the centroids of the clusters.
Intra-cluster distance: the distance between data vectors within a cluster.
Correct ratio Rc: the correct ratio of cluster numbers is defined by
(22)
where TCNC represents the number of times when the correct number of clusters was obtained by the algorithm and RT is the number of times where the algorithm was run.
AGFCM was compared with dynamic clustering particle swarm optimization (PSO) [40] and the standard GA (SGA). As the three algorithms used for comparison are stochastic in nature, we have undertaken 100 independent runs. The results have been stated in terms of the mean values for each couple (algorithm, data set).
The algorithms discussed in this section have been developed in a C++ language on a Core i7 PC, with a 2-GB Debian OS environment.
In Table 1, we report the mean number of classes found by the compared algorithms. The inter-cluster distance and intra-cluster distance obtained for the three algorithms are given in Tables 2 and 3.
Mean Number of Classes.
| Data set | SGA | PSO | AGFCM |
|---|---|---|---|
| Iris | 2.23±0.07 | 2.50±0.08 | 3.04±0.01 |
| Glass | 4.71±0.03 | 5.68±0.04 | 6.05±0.02 |
| Wine | 3.71±0.07 | 2.68±0.04 | 3.05±0.05 |
| Cancer | 2.22±0.06 | 3.01±0.04 | 2.04±0.04 |
Inter-cluster Distance.
| Data set | FCM | SGA | PSO | AGFCM |
|---|---|---|---|---|
| Iris | 2.05±0.05 | 2.105±0.08 | 2.412±0.09 | 2.598±0.15 |
| Glass | 840.20±6.15 | 898.20±9.15 | 869.42±8.01 | 853.12±3.08 |
| Wine | 2.15±0.15 | 2.20±0.15 | 2.42±0.06 | 3.12±0.04 |
| Cancer | 2.15±0.15 | 2.121±0.09 | 2.621±0.08 | 3.251±0.06 |
Intra-cluster Distance.
| Data set | FCM | SGA | PSO | AGFCM |
|---|---|---|---|---|
| Iris | 3.890±0.15 | 3.662±0.15 | 3.967±0.12 | 3.114±0.07 |
| Glass | 670.80±5.454 | 663.30±4.34 | 661.12±3.15 | 563.12±2.19 |
| Wine | 6.15±1.2 | 5.95±1.9 | 5.13±0.10 | 4.12±0.06 |
| Cancer | 5.01±0.25 | 4.984±0.25 | 4.538±0.10 | 4.037±0.08 |
The results in Tables 1–3 indicate that the AGFCM algorithm succeeds in obtaining the most appropriate number of classes over 100 runs. It has a good performance in terms of the inter-cluster distance and the intra-cluster distance. AGFCM obtains a better clustering of the data.
Table 4 gives the comparative data based on the correct ratio. It can be obviously deduced that the AGFCM algorithm remains clearly and consistently superior to its competitors.
Correct Ratio (%) of Cluster Numbers for Different Methods.
| Data set | SGA | PSO | AGFCM |
|---|---|---|---|
| Iris | 48 | 73 | 97 |
| Glass | 61 | 94 | 94 |
| Wine | 64 | 80 | 88 |
| Cancer | 44 | 81 | 96 |
The statistical results [18] of comparing AGFCM with SGA and PSO are given in Table 5. We used the one-tailed t-test with 58 degrees of freedom at a 0.05 level of significance, the changes in every τ generations. The notation used in Table 5 to compare each pair of algorithms is “+,” “++,” or “≈,” when the first algorithm is better than, significantly better than, or statistically equivalent to the second algorithm, respectively. Table 5 clearly demonstrates that the performance of AGFCM surpasses that of the other algorithms. When the change of population size N is small, the difference between the algorithms is not very meaningful. However, when the population size is large, AGFCM outperforms the other algorithms.
T-test Results of Comparing the Different Algorithms.
| T-test results | τ | Population size |
|||
|---|---|---|---|---|---|
| 50 | 100 | 200 | 500 | ||
| AGFCM-SGA | ≈ | + | + | ++ | |
| AGFCM-PSO | 10 | ≈ | + | + | + |
| AGFCM-SGA | + | + | ++ | ++ | |
| AGFCM-PSO | 50 | ≈ | ++ | ++ | ++ |
| AGFCM-SGA | + | + | ++ | ++ | |
| AGFCM-PSO | 100 | + | ++ | ++ | ++ |
| AGFCM-SGA | + | ++ | ++ | ++ | |
| AGFCM-PSO | 200 | + | + | ++ | ++ |
5 Conclusions
In this paper, we presented a hybrid GA for setting the appropriate number of clusters and determining initial cluster centroids for FCM. AGFCM uses two heuristic approaches, namely differential evolution and GSA, as a basis for genetic operators so as to exploit the best research areas and to explore other ones. The use of the MES, as a selection operator, controls the selection pressure. The experimental results on three real data sets indicate that the proposed algorithm improves the outcomes and outperforms two state-of-the-art clustering techniques. Future research may focus on integrating the automatic clustering scheme with the AGFCM algorithm for other metaheuristics such as PSO or differential evolution.
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