# Journal of Statistical Theory and Applications

Volume 19, Issue 1, March 2020, Pages 91 - 101

# Fuzzy C-Means Clustering Using Asymmetric Loss Function

Authors
1Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
2Telecommunication of Iran Company, Tehran, Iran
3School of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
Corresponding Author
Received 26 November 2018, Accepted 15 February 2019, Available Online 9 March 2020.
DOI
10.2991/jsta.d.200302.002How to use a DOI?
Keywords
Fuzzy C-Means clustering; LINEX loss function
Abstract

In this work, a fuzzy clustering algorithm is proposed based on the asymmetric loss function instead of the usual symmetric dissimilarities. Linear Exponential (LINEX) loss function is a commonly used asymmetric loss function, which is considered in this paper. We prove that the negative likelihood of an extreme value distribution is equal to LINEX loss function and clarify some of its advantages. Using such a loss function, the so-called LINEX Fuzzy C-Means algorithm is introduced. The introduced clustering method is compared with its crisp version and Fuzzy C-Means algorithms through a few real datasets as well as some simulated datasets.

Open Access

## 1. INTRODUCTION

Clustering is a method of creating groups of objects or clusters, in such a way that the objects in one cluster are very similar and the others are quite distinct. In hard clustering, one obtains a disjoint partitioning of the data such that, each data point belongs to exactly one of the partitions. In soft clustering, however, each data point has a certain probability (or possibility) of belonging to each of the partitions, which takes values between 0 and 1 [1]. One of the most widely used fuzzy clustering methods is the Fuzzy C-Means (FCM) algorithm, which introduced by Ruspini [2].

There are many techniques to group the observations into clusters, which use the loss functions to measure the dissimilarities between all pairs of observations such as Manhattan, Euclidean, Cosine, and Mahalanobis distances. A symmetric distance is useful for clustering when overestimating and underestimating of clusters are equally costly. That is, miss classifying object losses in two clusters have the same values. There is no common distance measure, which can be appropriate for all clustering applications. However, in some estimation problems, the use of an asymmetric loss function may be more suitable. To make our point clear, we give the following example:

Magic Gamma Telescope contains 19020 instances with 11 attributes, which use to depict high-energy gamma particles with the imaging technique in a ground-based aerial Cherenkov Gamma Telescope. It is split into two clusters, Gamma (signal), which includes 12332 data and Hadron (background) which includes 6688 data. The simple clustering accuracy is not significative for this data, because clustering a background event as the signal is worse than clustering a signal event as the background. The overestimation of a background event is preferable here. Therefore, using asymmetric dissimilarity measures helps us to classify data in the appropriate clusters to reduce the errors and risks resulting from the clustering. To present these data, we simplify the dataset artificially as follows:

Consider artificial data with one attribute and two classes that denoted by “circles” and “squares.” We define an error “e” that is the number of misclassified data and a penalized error “pe,” which is equal to “e” for the circles and “2e” for the squares. The scatterplot of 12 ordered data with respect to their values is plotted as a “Benchmark data” in Figure 1.

If we have no misclassified data in two clusters, both errors are zero (graph: top-right). However, when we have one misclassified data depends on it belong to which cluster the “pe” is equal to 1 for the circle and 2 for the square class (middle graphs). Note that misclassified data are bolded in the scatterplots. At the bottom of Figure 1, two cases of 2 misclassified data are plotted and two introduced errors are computed. We realize that symmetric loss function ignores such a penalty in the clustering result. Using asymmetric loss function in the estimation problem has a long history in the literature. Varian [3] and Berger [4] have considered asymmetric linear loss functions. In some estimation and prediction problems, the use of a symmetric loss function may be inappropriate. In practice, the overestimation and underestimation of the same magnitude of a parameter often have different losses, so the actual loss functions are asymmetric. Varian [3] proposed the asymmetric LINEX loss function in which, a given positive error might be more serious than a negative one of the same magnitude or vice versa [5]. Singh [6] used LINEX loss function to obtain an improved estimator for the mean in a negative exponential distribution. On the other hand, using an asymmetric loss function in clustering is used in the last decade. Soft clustering approach with Bregman divergences loss function was introduced in [7]. They proposed Bregman divergences as a dissimilarity measure for data drawn from the exponential family of distributions. Ahmadzadehgoli et al. [8] used the LINEX loss function as a dissimilarity measure in K-Means (KM) clustering. The main difference of LINEX loss function as the dissimilarity measure with respect to the introduced dissimilarity measure such as Bregman divergences in the KM clustering is the centers of clusters in the LINEX K-Means (LKM) algorithm are not the mean of their observations.

In this paper, we propose FCM clustering based on the LINEX loss function, which is called LINEX Fuzzy C-Means (LFCM). The paper is organized as follows: In Section 2, we recall the FCM algorithm. The LFCM clustering is introduced in Section 3. A characterization result and some advantages of LINEX loss function are proved and introduced in Section 4. Next, we evaluate the proposed algorithm by using some real datasets in Section 5. Section 6 dedicates to the robustness of the proposed algorithm. We conclude the paper in Section 7.

## 2. FUZZY C-MEANS

The FCM is a method of clustering, which allows one observation to belong to more than one cluster with a grade of membership ranging from 0 to 1. This method was introduced by Ruspini in 1970 [2], developed by Dunn in 1973 [9] and improved by Bezdek [10] in 1981. FCM has been widely used in cluster analysis, pattern recognition, and image processing. The FCM algorithm is more suited to data, which is more or less evenly distributed around the cluster centers. FCM partitions a given dataset X=x1,,xnp, where xk=(xk1,,xkp), k=1,,n, into c fuzzy subsets by minimizing the following objective function:

JFCMU,V;X=i=1ck=1nuikmxkvi2,(1)
where 1<c<n is the number of clusters, n is the sample size of data points. A matrix U=uikc×n represents fuzzy c-partition of X, where uik0,1 represents the grade membership of xk in the ith cluster that satisfies [1]
0uik1,i=1,,c,k=1,,n,(2a)
i=1cuik=1,k=1,,n,(2b)
0k=1nuikn,i=1,,c,(2c)

vi=vi1,,vip,i=1,,c is the ith class center, V is the set of cluster centers, . is the Euclidean norm between any measured data and the center vi, and m>1 is a weighting exponent of each uik. The parameter m is a quantity, which controls the clustering fuzziness.

The FCM algorithm is shown in the following steps [10,1,11]:

Choose initial centers vii=1,,c and 0<ε<1.

Compute the membership functions uik, assign the elements xk to the clusters, according to

uik=l=1cxkvi2xkvl22m11.
1. Update the cluster's center by the following expression:

vi=k=1nuikmxkk=1nuikm.

Let U(t) be the fuzzy c-partition in iteration t, if U(t)U(t1)<ε, then stop, otherwise, put t=t+1 and go to step 2.

## 3. LINEX FCM CLUSTERING

Many symmetric dissimilarity measures, such as squared Euclidean, Mahalanobis, Itakura-Saito, and relative entropy, have been used for clustering. When the overestimation and the underestimation of clusters are not of the same importance, an asymmetric dissimilarity measure is more appropriate. In this case, we propose the LINEX loss function as the dissimilarity measure in the FCM clustering algorithm.

The p-parameter LINEX loss function, proposed by Zellner [5], is

L(Δ)=j=1p(exp(ajΔj)ajΔj1),
where aj0,j=1,,p, and Δj=θ^jθj is the error in estimating θj using θ^j and Δ=Δ1,,Δp. When aj=1, the function is quite asymmetric with overestimation being more costly than underestimation. For all aj<0, when Δj<0, LΔ rises almost exponentially and almost linearly when Δj>0. For small values of |aj|, the function is almost symmetric and it is not far from a squared error loss function [12].

We want to state the LFCM algorithm, so consider the minimization equation (1), with the following LLINEX loss function instead of xkvi2 and denoted by JLINEX FCMU,V;X

LLINEX(xk,vi)=j=1pexpajxkjvijajxkjvij1,
JLINEXFCM(U,V;X)=i=1ck=1nuikmLLINEXxk,vi.

Now, we should minimize JLINEX FCMU,V;X, according to conditions (2a), (2b), and (2c). By the inception of [10], we solve the problem in the following two steps.

Step 1: Fix V=V^ and minimize JLINEX FCMU,V;X, it is minimized if and only if

uik=l=1cLLINEXxk,viLLINEXxk,vl1m11.(3)

Step 2: Fix U=U^, and then JLINEX FCMU^,V;X is minimized if and only if for each entity

vij=1ajlogk=1nuikmeajxkjk=1nuikm, for j=1,,p.(4)

The proofs of equations (3) and (4) are given in Appendices A and B, respectively. Therefore, to optimize JLINEX FCMU,V;X, the following two steps of the LFCM clustering algorithm should be modified in the FCM algorithm:

1. Compute the membership functions uik, according to relation (3).

2. Update the new cluster's centers vi, according to relation (4).

It is well known that the best estimator of the normal distribution location parameter can be obtained under squared error loss, and the estimator under absolute error loss function is not as well as it. The converse of this statement holds for Laplace distribution [14]. This observation has been extended to some classes of distributions. One of the most important extensions is the loss function for each member of the exponential family has a one to one correspondence with a Bregman divergences loss function [7]. This is a reason that some authors prefer to work with the negative log-likelihood function of data distribution instead of the loss function. That is, they prefer to find parametrically the best loss for parameters of a certain distribution. As well as the author's knowledge, this problem is not solved for the LINEX loss function. In the following, theoretically, we characterize a class of distribution, which LINEX gives the best results for the estimation of location parameters or cluster centers.

In the first step, we investigate that, is the LINEX an element of Bregman divergences loss functions? We answer this question in the first section indirectly. However, we can prove it by the contradiction method.

Consider the one-dimensional case. Assume that the LINEX loss function is a Bregman divergence. The loss function needs to satisfy the following equation for some ϕ [7]:

expaxyaxy1=ϕxϕyxydϕydy,x,yR,a>0.

Hence, ϕx=ax,ϕy=ay, which contradict to the following equation:

xydϕydy=expaxy1.

Therefore, the LINEX is not a Bregman divergences loss function.

To construct a density function, f, such that its negative log-likelihood is proportional to LINEX loss function, L, it should be as follows:

fx|μ,a=expLxμ/ga,μ,x,a>0,=1gaexpexpaxμaxμ1,(5)
where ga=expexpaxμaxμ1dx=e/a. Then, f can be simplified in the following form:
fx|μ,a=aexpaxμexpexpaxμ,μ,x,a>0,
which is well known as a generalized extreme value distribution, called Gumbel. The above discussion leads us to the following characterization lemma.

### Lemma 1.

Let f(x|μ,a) be the probability density function of Gumbel distribution with location μ and scale parameter 1a. Let LTμ be the LINEX loss function of real statistic T for estimating parameter μ with shape value a. Then f can be uniquely expressed as (5).

This result is interesting and important since extreme value distributions play a role similar to the normal distribution in the central limit theorem, i.e., they are convergent limits of extreme values of independent and identically distributed (iid) sequences. Therefore, the LINEX gives the best result with respect to the other loss functions in estimation or clustering as extreme values dataset.

### Remark 1.

The multi-parameter separable LINEX loss function of Zellner [5] given in Section 3 has a similar property as Lemma 1 for a vector of independent Gumbel distributions with different parameters.

In addition to the mentioned properties and advantages of an asymmetric loss function in a clustering problem in the first section, we can emphasize the following points: The most and the best extensions of FCM are limited to applying on a few flexible symmetric loss functions such as Minkowski and Entropy loss functions. Bregman divergence was the only well-known class of loss function, which is contained asymmetric loss and used in fuzzy clustering. The main difference between this class with the LINEX loss, in a clustering problem, is the characterization of two different classes of distributions. On the other hand, the cluster centers in the LFCM algorithm are not a weighted mean of the data in each cluster, recall equation (4).

We continue this section with a simulation study to see the performance of the proposed clustering method with respect to the KM, FCM, and LKM clustering algorithms. We consider clustering problem of two-class datasets with two independent variables. The data proportion of each class in all datasets is equal.

Three datasets of size 200 are generated from the two-component bivariate mixture of Normal, Laplace, and Gumbel distributions with the same scale and location parameters (0, 0) and (3, 3) for each class respectively, see Figure 2. We would like to show the effect of the scale parameter, 1a, on the clustering precision of the three generated datasets. We plot graphs of the miss-classified percentage of data (Error %) with respect to the scale parameter in the left column of Figure 3. Each value is computed as the average of 2000 simulation iterations. To compare the robustness of the algorithms, the standard deviation (StDev) of miss-classified percentage are also plotted in the right column of Figure 3. The value of a for clustering Normal and Laplace datasets is fixed to 0.0001 for the algorithms LKM and LFCM. However, for the Gumbel dataset, based on the advice of Lemma 1, the value of a is proportional to the scale parameter of generating datasets. The parameter, m, for the two fuzzy algorithms, FCM and LFCM, is considered the same value of 2 in all simulations.

We report the simulation result for the scale parameter in [0.4, 1.6], see Figure 3. The first observation from the graphs of this figure is the accuracy similarity of fuzzy algorithms and the similarity of non-fuzzy algorithms. That is, FCM and LFCM are better than KM and LKM in both of error percentage (left column of Figure 3) and their StDev (right column of Figure 3). In the Normal and Laplace datasets, the miss-classified percentage of FCM and LFCM are very close. However, in the Gumbel dataset LFCM is the best with the aid of Lemma 1. In overall, we can conclude that, in this experiment, LFCM is better than FCM. Also, FCM is better than LKM, which is better than KM.

In the next section, we study the preference of mentioned clustering methods through real datasets. We conclude this section with two practical remarks.

### Remark 2.

The Gumbel distributions in Lemma 1 and Figure 2 are skewed to the left. If the data have a right heavy tail, the LINEX loss function is appropriate for the negative of data.

### Remark 3.

In a real problem, the unknown parameter a for each variable can be estimated or find it through a cross-validation procedure. Furthermore, a small value, near zero, for a is useful in the case of no underestimation or overestimation preference in clustering.

## 5. EVALUATION

In this section, we employ clustering accuracy as a measure for evaluating the clustering results. Clustering accuracy is an external measure, which computes the percentage of the correctly classified data points according to the class labels [15].

We use the Normalized Variation Information (NVI) and Davies–Bouldin (DB) criteria. We compare the results with FCM and LKM clustering algorithms to check the proposed algorithm performance.

NVI is an external criterion, which is used to compare the clustering performance across original labels of datasets. If it falls in the range of 0,1, the clustering is considered high quality [16]. DB is an internal index, which measures the ratio of the sum of within-cluster scatters between cluster separations. It shows a better result when it is close to zero, which indicates that the clusters are more compact and are separated better [17,18].

In this part, we use some real datasets, which are in the UC Irvine Machine Learning Repository [19] to verify the results of LFCM, FCM, and LKM. The datasets are listed in the following:

1. The Iris dataset contains 150 instances; each one consists of four features and they are grouped into three classes Setosa, Versicolour, and Virginica with 50 instances. Each class refers to a type of Iris plant.

2. The Wine dataset is the results of a chemical analysis of wines, which is derived from three different cultivars. It contains 178 instances; each one consists of 13 features and is grouped into three classes with 62, 69, and 47 instances.

3. The Seeds dataset is the examined group kernels containing 210 instances with seven features and is belonged to three varieties of wheat: Kama, Rosa, and Canadian with 70 elements.

4. The Haberman's Survival dataset contains 306 instances; each with three features and is grouped into two classes: the patient who is alive five years or longer and the patient died before five years.

5. The MAGIC Gamma Telescope contains 19020 instances; each one consists of 11 features and is grouped into two classes, Gamma (signal) with 12332 instances and Hadron (background) with 6688 instances.

Now, we want to evaluate the results on the real datasets. We run the algorithms 100 times and compute the accuracy, NVI, and DB each time for the first experiment. The results are given in Table 1. In all computations, we fix m=2 and ε=0.01.

Dataset Evaluation Index FCM a=103
LKM LFCM
Iris Accuracy 0.8797 0.8487 0.8797
NVI 0.3895 0.4950 0.4005
DB 0.4640 0.5390 0.4997
Wine Accuracy 0.7136 0.7212 0.6619
NVI 0.7332 0.7122 0.7690
DB 0.4872 0.4871 0.4723
Seeds Accuracy 0.8744 0.8714 0.8744
NVI 0.4675 0.4495 0.4675
DB 0.6788 0.8558 0.6788

FCM, Fuzzy C-Means; LKM, LINEX K-Means; LFCM, LINEX Fuzzy C-Means; NVI, Normalized Variation Information; DB, Davies–Bouldin.

Table 1

The comparison between FCM, LKM, and LFCM in terms of accuracy, NVI and DB values in real datasets. The best values are bolded.

The results in Table 1 show that the LFCM algorithm performance is not so different from the FCM for the values of the parameter a close to zero, but it is more accurate in comparison with the LKM algorithm except in one case.

Recall the example in the first section. In the Magic Gamma Telescope dataset, underestimation of backgrounds has more penalty with respect to the signals. Therefore, calculated accuracy in Table 1 are not fair for comparing clustering algorithms. In the next experiment, we try to show the efficiency of the proposed algorithm when a penalty considered for underestimation. We assume that loss of a background classified as signal be two units and the converse be one unit. Then the loss of miss-classified data with considering a penalty, which is computed as follows:

Penalized loss=n1×1+n2×219020,
where the size of data is 19020, n1 and n2 are the numbers of data with losses equal to “class 1” and “class 2” respectively. Now we run the algorithms 50 times to report the average values of all evaluation indices similar to Table 1. The results of this experiment presented in Table 2. We choose the value a>0 in steps of 0.1 according to the NVI and DB values to find the best value of a. The results in Table 2 show that the LFCM algorithm performance is more accurate than the LKM and FCM. We see that LFCM has a minimum loss considering on the penalty.

Dataset Evaluation Index FCM LKM LFCM
Haberman's survival Accuracy 0.5196 0.7164 0.7581
NVI 1.0000 0.9998 0.9579
a=0.5 DB 0.9652 0.9013 0.9579
Penalize loss 0.6274 0.4313 0.4151
Magic gamma telescope Accuracy 0.6113 0.6832 0.7015
NVI 0.9949 0.1230 0.9620
a=0.1 DB 1.3770 0.8920 1.8950
Penalize loss 0.6130 0.7967 0.5413

FCM, Fuzzy C-Means; LKM, LINEX K-Means; LFCM, LINEX Fuzzy C-Means; NVI, Normalized Variation Information; DB, Davies–Bouldin.

Table 2

The comparison between FCM, LKM, and LFCM in terms of accuracy, NVI and DB values in real datasets with overestimation. The best values are bolded.

## 6. COMPLEXITY AND ROBUSTNESS

The main difference between KM and LKM is due to the center of a cluster. It is well known that, in KM algorithm, the centers of clusters are the means of clusters. However, for the LKM the centers deviate from the mean of clusters, based on the underestimating or overestimating strategy. This is the case for the FCM and LFCM. In step 3 of the FCM algorithm, the centers of clusters are recalled, which are the weighted means of the corresponding classes. We calculate the centers for the LFCM in (4), see Appendix B. By this introduction, we see that the LKM or LFCM just modified centers based on a new criterion. Therefore, the complexity of the proposed algorithm is similar to the FCM.

To show the robustness of the proposed algorithm, we consider five criteria or parameters and try to explain the behavior of the proposed algorithm through an experiment, which is classified in Table 2. The results confirm the similarity of LFCM and FCM for the convergence behavior, robustness, and sensitivity of the mentioned criteria in Table 3.

Robustness Experiment Result Conclusion
Convergence behavior We repeat the LFCM three times for simulated and real datasets that introduced in Section 4. Based on the stopping error U(t)U(t1) the convergence is verified. Reach to fixed points of the final centers. The recorded maximum number of iterations were obtained 18 and 47 for the simulated and real datasets. This experiment confirms theoretical results.
Scalability The convergence behaviors of the LFCM algorithm for simulated data with sizes 200, 2000, and 20000 are considered. We have observed that the algorithm converges before iteration 40. This confirms the computational complexity of LFCM in this range of data size.
Influence of the fuzziness parameter We choose different values of m=2,4,6,8 and check convergence of LFCM. Based on the interior and exterior criteria for evaluation, the best value of m is determined. Based on each evaluation criterion, we have a better m which is not necessarily equal. However, their results have not been the main differences. We should choose an appropriate evaluation criterion before finding the best value of m.
Sensitivity to the stop condition We choose different values of stopping error threshold ε=0.01,0.03,0.05,0.1. The evaluation criteria are computed for each ε. The performance of the algorithm in some datasets has not been affected by the values of ε. Usually smaller ε gives a better evaluation criterion value or clustering result. The range of (0.01, 0.1) for ε gives good clustering results.
Sensitivity to initialization We run the algorithm 100 times with random seed numbers. The convergence of evaluation criteria is considered. We consider accuracy, NVI, and DB criteria. All criteria are convergent. LFCM is robust with respect to the initialization.

LFCM, LINEX Fuzzy C-Means; NVI, Normalized Variation Information; DB, Davies–Bouldin.

Table 3

Results of investigating five robustness criteria, experiment, result, and its conclusion for the proposed algorithm.

## 7. CONCLUSION

The type of distance based on loss function has an important role in the clustering analysis. When the overestimating or underestimating has the same magnitude, the symmetric loss function is a suitable one. However, when they have different significances, an asymmetric loss function should be employed. In this paper, the well-known LINEX loss function, as the asymmetric dissimilarity distance, was used to develop a new fuzzy clustering method. We stated and proved a characterization lemma to show that LFCM give an optimal result for datasets from Gumbel distribution, which is an extreme value distribution. The performance of the proposed algorithm was presented in a simulation study. Some real datasets were used to investigate the results of the LFCM and to compare with the LKM and FCM clustering algorithms. The comparisons were done based on NVI and DB criteria. The accuracy of the LFCM algorithm was good and depended on the parameter a which is determined according to the lower values of NVI and DB.

## CONFLICT OF INTEREST

Authors have no conflicts of interest to declare.

## ACKNOWLEDGMENTS

The authors would like to thank the editor and the anonymous reviewers for their useful comments and suggestions that improve the manuscript, especially in Section 5. We also thank Prof. K. Shafie for his comments on the revised part of the manuscript.

## APPENDIX A

The object function JLINEX FCMU,V;X with constraint conditions (2a), (2b), and (2c) can be solved by using the Lagrangian multiplier as follows:

To find minJLINEX FCMU,V;X it is sufficient to minimize the following inner sum for fixed k:

i=1cuikmLLINEX FCMxk,vi,xk,vip.

Put Lik=LLINEX FCMxk,vi.

Let B=uk=u1k,,uckc|i=1cuik=1,0uik1, and

guk=i=1cuikmLikxk,vi.

The Lagrangian of guk is defined as follows:

Fλ,uk=i=1cuikmLikλi=1cuik1,

λ,uk is stationary for F only if λ,ukFλ,uk=0, 0c that yields to

Fλλ,uk=i=1cuik1=0,(A.1)
Fuikλ,uk=muikm1Likλ=0.(A.2)

From (A.2)

uik=λmLik1m1=λm1m11Lik1m1.(A.3)

Then,

ulk=λmLlk1m1=λm1m11Llk1m1,ulkB.

From (A.1)

l=1culk=l=1cλm1m11Llk1m1=λm1m1l=1c1Llk1m1=1.

Hence,

λm1m1=1l=1c1Llk1m1.(A.4)

Substitute (A.4) in (A.3), we obtain

uik=1l=1c1Llk1m11Lik1m1=1l=1cLikLlk1m1=l=1cLikLlk1m11=l=1cLLINEX FCMxk,viLLINEX FCMxk,vl1m11.

## APPENDIX B

(Proof of minimizing JLINEX FCMU,V;X)

To prove, it is sufficient to minimize the following inner sum for fixed i

k=1nuikmeajxkjvijajxkjvij1.

To do this, we differentiate with respect to vij, and the result is obtained.

k=1nuikmeajxkjajvijajk=1nuikmxkj+ajk=1nuikmvijk=1nuikmvij=0
ajk=1nuikmeajxkjajvij+ajk=1nuikm=0eajvij=ajk=1nuikmajk=1nuikmeajxkjvij=1ajlogk=1nuikmeajxkjk=1nuikm.

## REFERENCES

3.H.R. Varian, S.E. Fienberg and A. Zellner (editors), Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, North-Holland Pub. Co., Amsterdam, Netherlands, 1975, pp. 195-208.
6.B.K. Singh, Int. J. Soft Comput. Math. Control, Vol. 2, 2013, pp. 27-44. https://wireilla.com/ns/maths/Papers/2113ijscmc04.pdf
7.A. Banerjee, S. Merugu, I.S. Dhillon, and J. Ghosh, J. Mach. Learn. Res., Vol. 6, 2005, pp. 1705-1749. http://www.jmlr.org/papers/volume6/banerjee05b/banerjee05b.pdf
16.N.X. Vinh, J. Epps, and J. Bailey, J. Mach. Learn. Res., Vol. 11, 2010, pp. 2837-2854. http://jmlr.csail.mit.edu/papers/volume11/vinh10a/vinh10a.pdf
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
19 - 1
Pages
91 - 101
Publication Date
2020/03/09
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.200302.002How to use a DOI?
Open Access

TY  - JOUR
AU  - Israa Abdzaid Atiyah
AU  - S. Mahmoud Taheri
PY  - 2020
DA  - 2020/03/09
TI  - Fuzzy C-Means Clustering Using Asymmetric Loss Function
JO  - Journal of Statistical Theory and Applications
SP  - 91
EP  - 101
VL  - 19
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200302.002
DO  - 10.2991/jsta.d.200302.002
ID  - Atiyah2020
ER  -