Performance Analysis of a Novel IT2 FCMAlgorithm

Shashank Anil Huddedar∗, Mayank Kagliwal†, Badrinath Singhal‡, and Frank Chung-Hoon Rhee, Member, IEEE§∗Department of Computer Science and Engineering

Indian Institute of Technology Guwahati, Guwahati, India 781039Email: huddedar@iitg.ac.in†Department of Mathematics

Indian Institute of Technology Guwahati, Guwahati, India 781039Email: m.kagliwal@iitg.ac.in

‡Department of Electronics and Electrical EngineeringIndian Institute of Technology Guwahati, Guwahati, India 781039

Email: badrinath@iitg.ac.in§Department of Electronic Engineering

Hanyang University, Korea 15588Email: frhee@fuzzy.hanyang.ac.kr, Telephone: +82.31.400.5296, Fax: +82.31.436.8152

Abstract—In this paper, we propose a novel interval type-2(IT2) fuzzy clustering algorithm by incorporating a speed up typereduction algorithm. In order to illustrate our proposed method,embedded lines and planes that are associated with the IT2 fuzzymembership functions (MFs) are confined to 2-dimensional (2-D)space for visualization purposes. The original IT2 fuzzy C-means(FCM) algorithm uses the Karnik-Mendel (KM) algorithm as apart of its type reduction procedure where computation of thecentroid is achieved by iterating each dimension of the patternsets separately. This ignores the possible correlation amongthe multiple dimensions and can result in high computationalcomplexity. Our proposed algorithm considers multidimensionalpattern sets jointly and estimates the centroid at comparablecosts. Finally, experiments are performed on several pattern setsto show the validity of our proposed method.

I. INTRODUCTION

Many pattern recognition tasks may involve imperfectknowledge or information about the pattern sets. Althoughtype-1 (T1) fuzzy sets (FSs) are capable of handling theuncertainties present in pattern sets effectively, they lack theability to perform the same when there exist uncertainties invarious parameters such as distance measure, fuzzy degree,and membership values [1]-[4]. On the other hand, type-2 (T2)FSs allow us to model uncertainties more effectively [5]-[8].There exists wide range of applications where T2 FSs havebeen successfully used such as robotics [9], clustering [10],and classification of coded video streams [11], to name a few.

Out of the many proposed T2 fuzzy clustering algorithms,the interval T2 (IT2) fuzzy C-means clustering (FCM) [12]algorithm has gained noticeable attention due to its per-formance. One of the essential steps involved in the IT2FCM algorithm are type reduction and defuzzification. Theoriginal Karnik-Mendel (KM) algorithm [13] and variantssuch as the enhanced KM (EKM) [14], iterative algorithmwith stopping condition (IASC) [15], and enhanced IASC

(EIASC) [16] algorithms used in type reduction are applicablefor multidimensional pattern sets where computation of thecentroid is achieved by iterating each dimension of the patternsets separately (i.e., processed in 1-dimension (1-D)). Thesemethods ignore the possible correlation among the multipledimensions and can result in high computational complexity.By considering multidimensional pattern sets jointly, the esti-mates of the centroid may be computed with better accuracyat comparable costs. In this paper, we use the concept ofembedded lines and planes as explained in [17]. We integratethe algorithm proposed in [17], which we call as the modifiedKM algorithm, in the original IT2 FCM algorithm to give riseto a novel modified IT2 FCM algorithm. In order to visualizeand effectively illustrate our proposed method, we restrict ourstudy to 2-D pattern sets [18].

In Section II, we formally present our proposed modifiedIT2 FCM algorithm and describe the type reduction anddefuzzification processes in detail. In Section III, we comparecomplexities of the original KM and modified KM algorithms.In Section IV, we give some experimental results on patternsets to show the validity of our proposed method. Finally inSection V, we conclude the paper by stating the areas wherethe modified IT2 FCM algorithm will be more effective thanboth the FCM and the original IT2 FCM algorithms.

II. PROPOSED MODIFIED IT2 FCM ALGORITHM

In this section, we describe a modified IT2 FCM algo-rithm that incorporates a speed up type reduction algorithm.The fundamental difference between original IT2 FCM andmodified IT2 FCM algorithm is in the type reduction anddefuzzification steps. In the following subsections, we focus ontype reduction and defuzzification steps used in the modifiedIT2 FCM algorithm that give rise to final crisp center.

978-1-5090-6020-7/18/$31.00 ©2018 IEEE

A. Description of the modified KM algorithm

A 2-D T2 FS, denoted as A, is a bivariate function on theCartesian product µA : X × [0, 1] → [0, 1], where X ⊂ R2

and its membership function is denoted by µA(x, u) wherex ∈ X and u ∈ U ⊆ [0, 1]. We can represent the T2 FS A as

A = {((x, u), µA(x, u)) : ∀x ∈ X,∀u ∈ U ⊆ [0, 1]}. (1)

The KM algorithm is an iterative algorithm which is used tofind the centroid of a 1-D T2 FS. When the KM algorithmis used to obtain the centroid of a 2-D T2 FS, we obtainbounds on the centroid, namely left and right centroid ofthe corresponding embedded sets for each dimension. Thesecentroids are computed as

cl,d(L) =

L∑i=1

xi,d µA(xi) +N∑

i=L+1

xi,d µA(xi)

L∑i=1

µA(xi) +N∑

i=L+1

µA(xi)

(2)

and

cr,d(R) =

R∑i=1

xi,d µA(xi) +

N∑i=R+1

xi,d µA(xi)

R∑i=1

µA(xi) +

N∑i=R+1

µA(xi)

, (3)

where L and R are the respective switch points. The uppermembership function (UMF) and lower membership function(LMF) are µA and µ

A, respectively, for xi = (xi,1, xi,2),

where d ∈ {1, 2}.Although the KM algorithm provides a good approximation

of the left cl(L) (cl(L) = {cl,1(L), cl,2(L)}) and right cr(R)(cr(R) = {cr,1(R), cr,2(R)}) centroid bounds, it does notconsider the correlation between the multiple dimensions ofpattern sets. In addition, iterating each dimension of the patternsets can involve undesirable computations. To remedy thisdrawback, a new KM (modified KM) algorithm was proprosedin [17].

The main aim behind the modified KM algorithm is tovisually estimate a centroid boundary of 2-D IT2 FSs. First,we fix θ ∈ [0, π] such that a line creating an angle π

2 + θwith the dimensional feature axis (x1-axis) is used as areference axis. Then, instead of iterating each dimension ofpattern set, we project the entire multidimensional domain toa single dimension. This dimensionality reduction is achievedby sorting the projections of points on the reference axis.The modified KM algorithm is analogous to the original KMalgorithm where the reference axis corresponds to a referencepoint, namely the origin and the projection of each point onthe reference axis corresponds to the distance of the pointsfrom the origin. The embedded set obtained as a result ofdimensionality reduction is given as

Aθ =∑

x∈X

∑u∈µA(x)

1

/(x ·[cos(θ)sin(θ)

], u). (4)

Then, we apply the KM algorithm on this 1-D T2 FS. As aresult, convergence of the modified KM algorithm provides

(a)

(b)

Fig. 1: Embedded set corresponding to an embedded-curvefor direction θ: (a) side view and (b) top view

two centroids namely, cl,θ and cr,θ. The embedded linescorresponding to these centroids are defined as

Lθ(x) ≡ x ·[cos(θ)sin(θ)

]− cr,θ (5)

and

Lπ+θ(x) ≡ x ·[cos(π + θ)sin(π + θ)

]+ cl,θ. (6)

The embedded line corresponding to cr,θ, i.e., Lθ(x) is shownin Fig. 1 and the corresponding embedded set is given by

Aθ =∑

x:Lθ(x)≤0µA(x)/

x +∑

x:Lθ(x)>0µA(x)

/x,

(7)where µA and µ

Aare the respective UMF and LMF of the IT2

FS (see Fig. 2). The centroid of this embedded set is obtainedby

c(Aθ) =

∑x∈X µAθ

(x) · x∑x∈X µAθ

(x). (8)

The above procedure is repeated for a predetermined num-ber of distinct θ, where θ takes discrete values in [0, π]. Byconnecting the centroids, we obtain the centroid boundarywhich consists of all the points of the set c(Aθ). A step bystep flow of the modified KM algorithm is given in Fig. 3.

B. Defuzzificaton of the centroid boundary

The process of defuzzification involves estimating a crispcenter from a T1 FS. In the current discussion, we defuzzify

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Fig. 2: Footprint of Uncertainity (FOU)

the left and right centroid boundaries separately to give finalcentroid bounds. These centroid bounds are further defuzzifiedto give rise to final crisp center.

From the above process of type reduction, we obtain twocentroid boundary sets, namely Sl and Sr which are given as

Sl = {c(Aθ1), c(Aθ2), ..., c(Aθn)} (9)

andSr = {c(Aπ+θ1), c(Aπ+θ2), ..., c(Aπ+θn)}. (10)

Here, θ is discritized into n different values in [0, π]. Asboth centroid boundaries consist of interval T1 (IT1) FSs, wedefuzzify them using

vl =

n∑i=1

c(Aθi)

n(11)

and

vr =

n∑i=1

c(Aπ+θi)

n. (12)

Finally, the centroid of IT2 FS is obtained by

v =vl + vr

2. (13)

We present the detailed flow of the modified IT2 FCMalgorithm as shown in Fig. 4.

C. Pseudo Code and Flowchart

In the modified KM algorithm, data represents the 2-D T2 FS under investigation and Ul (Uh) represent LMF(UMF). In addition, Vin represents the initial center position,Vold represents the previous center position, Vnew representsthe initial center position, C represents the total number ofclusters, Um1 and Um2 represent the respective estimated FCM

membership using fuzzifier values m1 and m2 [12] which aregiven by

Um1 =1

C∑k=1

(||xi−cj ||||xi−ck|| )

2m1−1

(14)

and

Um2=

1C∑k=1

(||xi−cj ||||xi−ck|| )

2m2−1

. (15)

Algorithm 1 Modified KM Algorithm

1: procedure MODIFIED KM(data, Ul, Uh)2: sum← 03: Set directions, dir = {d1, d2, ..., dn}4: for each direction di in dir do5: Sort projections of data on direction, di.6: Sort Uhi and Uli in terms of updated index of data.7: center ← KM(data, Uli , Uhi)8: sum← sum+ center

9: end for10: Vfinal ← sum/n11: return Vfinal

Fig. 3: Modified KM Algorithm

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Algorithm 2 Modified IT2 FCM Algorithm

1: procedure MODIFIED IT2 FCM(data, Vin,m1,m2)2: Vnew ← Vin3: do4: Vold ← Vnew5: Um1

← 1C∑k=1

(||xi−cj ||||xi−ck||

)2

m1−1

6: Um2 ← 1C∑k=1

(||xi−cj ||||xi−ck||

)2

m2−1

7: for i = 1 to clusters do8: for j = 1 to patterns do9: Uh(i, j)← max (Um1

(i, j), Um2(i, j))

10: Ul(i, j)← min (Um1(i, j), Um2

(i, j))

11: end for12: end for13: Vnew ←MODIFIED KM(data, Ul, Uh)14: while Vold 6= Vnew

Fig. 4: Modified IT2 FCM Algorithm

III. ANALYSIS OF COMPUTATIONAL COMPLEXITY

The significant difference in the original IT2 FCM andmodified IT2 FCM algorithms is in the type reduction step.Hence, we focus our discussion on the comparative study of

the existing KM and modified KM algorithms. The large over-head for the existing KM algorithm is mainly constituted inthe sorting of the pattern sets in each dimension. We overcomethis issue by applying dimensionality reduction of the patternsets by sorting their projections based on the reference axisand use the KM algorithm on this 1-D T2 FS to obtain thefinal centroids. It is observed that the major computationalcost involved in the case of the modified KM algorithm isdue the number of directions. Considering the number ofdirections as a hyperparameter (d), it can be adjusted basedon the sparsity of the pattern sets in multidimensional space toachieve better results. If the total number of pattern sets are Nand total number of dimensions are m, then the complexity ofthe KM algorithm is O(m · N log(N)). On the other hand,the complexity of the proposed modified KM algorithm isO(d·N log(N)). Therefore, by controlling the hyperparameterd, we can achieve improved results at comparable costs.

IV. EXPERIMENTAL RESULTS

A. Visualization of the centroid region

In the case of the original IT2 FCM algorithm, we canvisualize the uncertainty of 2-D pattern sets by plotting thecl and cr in 2-D. By involving only two sets of points, weare able to visualize centroid boundary as a rectangle only.In case of the modified IT2 FCM algorithm, we plot the T1FSs of centroids considering each direction. The uncertaintyregion results in a polygonal shape. To verify this claim, weperform the following experiment.

We consider a 2-D pattern set that consists of three uni-formly distributed gaussian pattern sets with a particular meanand standard deviation. The modified IT2 FCM algorithm wasimplemented with the generated data and pairs of cl and crwere plotted for every direction.

For the coordinate directions, an approximately rectangularregion was found to be the centroid region. This experimentwas reconducted for a large number of directions and theshape of the region was found to tend towards an ellipse.The intermediate embedded line, embedded plane and themembership functions can be visualized as in Fig. 5. Fig. 6,Fig. 7, and Fig. 8 show the centroid region when numberof directions considered in modified KM algorithm are 2, 3,and 180 respectively. Plots to the left in Fig. 6, Fig. 7, andFig. 8 are magnified to emphasis the shape of the boundaries.In addition, shape of the centroid boundary represents thedistribution of the pattern sets in corresponding cluster. Thishelps in visualizing the centroid boundary better than theoriginal IT2 FCM algorithm.

B. Estimation of positioning error of cluster centers for theFCM, original IT2 FCM and modified IT2 FCM algorithms

In case of the modified IT2 FCM algorithm, we first find theT1 FS of the centroids for each direction. We get the final crispcenter by defuzzifying this set. So, we can consider the centerobtained by the modified IT2 FCM algorithm as the “center ofcenters.” On the other hand, the original IT2 FCM algorithmgives only one center instead of a T1 FS. By introducing noise

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(a)

(b)

(c)

Fig. 5: Visualization of (a) embedded line, (b) embeddedplane, and (c) upper and lower membership functions

-0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6

0.5

0.6

Fig. 6: Visualization of centroid boundary for 2 directions

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6

0.5

0.6

Fig. 7: Visualization of centroid boundary for 3 directions

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6

0.5

0.6

Fig. 8: Visualization of centroid boundary for 180 directions

in pattern sets, we check for the robustness of the algorithm.The center obtained by the modified IT2 FCM algorithm isaveraged over a set of directions, as a result of which we canexpect this algorithm to be more robust compared to the FCMand original IT2 FCM algorithms. To verify this claim, weperform the following experiment.

We consider two non-overlapping 2-D clusters by sampling150 points from two uniform distributions with predeterminedmean and standard deviation as shown in Fig. 9 [21]. TheFCM (m = 2.5), original IT2 FCM (m1 = 2, m2 = 3), andmodified IT2 FCM (m1 = 2, m2 = 3) algorithms were set tothe task of estimating the center of the distributions. To getthe maximum clustering result number of directions were setto 180, where θ ∈ [0, π] with resolution of 1 radian. In the

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0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(a)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(b)

Fig. 9: (a) Clean and (b) Noisy pattern sets

0 20 40 60 80 100 120 140 160

No. of noise points

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Err

or

in p

osi

tio

n o

f ce

nte

r

FCM

Modified IT2 FCM

IT2 FCM

Fig. 10: Error comparison between the centers obtained fromFCM, original IT2 FCM, and modified IT2 FCM algorithms

next step, for each iteration, 10 uniformly distributed pointswere added to the distribution. This process was continued till16 iterations (160 noise points). The experiment was repeatedseveral times to reduce the variance of the pattern set. The usedquality measure was a sum of position errors estimated by theFCM, original IT2 FCM, and modified IT2 FCM algorithms.Fig. 10 shows calculated errors for different number of noisepoints present in the pattern set.

Based on Fig. 10, we can draw the following observations.It is visible that the error in center positioning by the originalIT2 FCM and modified IT2 FCM algorithms are similarwhen noise present in the pattern sets is low. On the otherhand, the FCM algorithm was observed to have high errorin center positioning at every stage of the experiment. Asnoise increases, the error in center positioning is less in themodified IT2 FCM algorithm but is more in case of the originalIT2 FCM algorithm. Hence, significant improvement in centerpositioning can be obtained by using the modified IT2 FCMalgorithm when pattern set contains high noise.

C. Performance analysis of the FCM, original IT2 FCM andmodified IT2 FCM algorithms for overlapping pattern sets

The main goal of this analysis is to compare performancesof the FCM, original IT2 FCM and modified IT2 FCM

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

(a)

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

(b)

Fig. 11: Overlapping pattern sets

algorithms in the task of uncertain pattern recognition in caseof overlapping pattern sets. From the previous experiments,results indicate that on average, the modified IT2 FCM algo-rithm can provide better center as compared to that given byboth the FCM and original IT2 FCM algorithms. Therefore,by applying our proposed modified IT2 FCM algorithm to thetask of uncertain pattern recognition, we expect this algorithmto give a better performance than both the FCM and originalIT2 FCM algorithms. To verify this claim, we perform thefollowing experiment.

We draw 400 points from a uniform distribution and createtwo partially overlapping pattern sets consisting of 100 pointsin one cluster and 300 points in the other cluster. The twoclusters vary in size and density. 25 test cases were consideredin this experiment. Starting with 100 points in the smallercluster and 300 points in the larger cluster as shown inFig. 11(a), we repeatedly remove a fixed number (10 as inour experiment) of data points from the larger cluster andintroduce the same number of points in the smaller clusterin subsequent test cases. Removal and introduction of pointswas done randomly. Fig. 11(b) shows the final distribution ofpattern sets. In this procedure, the uncertainty in the patternrecognition task is due to cluster overlap and difference incluster size and density. The FCM, original IT2 FCM, andmodified IT2 FCM algorithms were applied to the task ofpattern recognition with fuzzifier values, m = 2.5, m1 = 2,and m2 = 3 . This experiment was repeated 30 times to reducevariance of the pattern set. To optimize the performance, weconsidered 180 directions in case of the modified IT2 FCMalgorithm, where θ ∈ [0, π] with resolution of 1 radian.

The results were found to be consistent as mentioned in[21]. The proposed algorithm was observed to perform betterthan both the FCM and original IT2 FCM algorithms whenoverlapping was low. After a certain threshold of overlapping,all the algorithms performed almost equally which is evidentfrom Fig. 12. Hence, it can be concluded that on an average,the modified IT2 FCM algorithm shows better recognition ratethan both the FCM and original IT2 FCM algorithms in thecase of overlapping pattern sets.

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0 5 10 15 20 25

Test cases #

0

20

40

60

80

100

Perf

orm

nace %

IT2 FCM

Modified IT2 FCM

FCM

(a)

0 5 10 15 20 25

Test cases #

50

56

62

68

74

80

Perf

orm

nace %

IT2 FCM

Modified IT2 FCM

FCM

(b)

Fig. 12: Performance comparison between the FCM, originalIT2 FCM, and modified IT2 FCM algorithms: (a) Normal

view and (b) Magnified view

V. CONCLUSION

In this paper, we proposed a novel IT2 FCM algorithmwhich utilized the concept of embedded lines and planes in 2-D space to determine the switch points for a particular cluster.Different directions of the reference axis were used to find theswitch points in respective directions. From the experimentalresults, we conclude that our proposed modified IT2 FCMalgorithm• provided the surrounding boundary inside which the

actual center was located, whereas the original IT2 FCMalgorithm gave only the left and right centroid bounds.In addition, shape of the centroid boundary representsthe distribution of the pattern sets in the correspondingcluster.

• performed better than both the FCM and original IT2FCM algorithms on pattern sets containing noise.

• gave better recognition rate on average when overlappingpattern sets were considered as compared to the FCM andoriginal IT2 FCM algorithms.

Hence, it can be concluded that our proposed modified IT2FCM is more robust to noisy and overlapping pattern sets as

compared to both the FCM and original IT2 FCM algorithms.In the current implementation of the proposed modified

KM algorithm, the directions are chosen uniformly from thediscritized domain of θ, where θ ∈ [0, π]. For further study, wewill try to tune this hyperparameter, θ based on the domainof the pattern sets to improve the complexity even further.In addition, rigorous study can be performed for finding theset of directions with minimum cardinality to obtain the bestpossible clustering result.

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