NovelTwo-DimensionalVisualizationApproachesfor...

Research ArticleNovel Two-Dimensional Visualization Approaches forMultivariate Centroids of Clustering Algorithms

Yunus Dogan 1 Feristah Dalkılıccedil 1 Derya Birant1 Recep Alp Kut1 and Reyat Yılmaz2

1Department of Computer Engineering Faculty of Engineering Dokuz Eylul University Izmir Turkey2Department of Electrical and Electronics Engineering Faculty of Engineering Dokuz Eylul University Izmir Turkey

Correspondence should be addressed to Yunus Dogan yunuscsdeuedutr

Received 8 January 2018 Revised 23 June 2018 Accepted 9 July 2018 Published 8 August 2018

Academic Editor Jose E Labra

Copyright copy 2018 Yunus Dogan et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)e dimensionality reduction and visualization problems associated with multivariate centroids obtained by clustering al-gorithms are addressed in this paper Two approaches are used in the literature for the solution of such problems specificallythe self-organizing map (SOM) approach and mapping selected two features manually (MS2Fs) In addition principlecomponent analysis (PCA) was evaluated as a component for solving this problem on supervised datasets Each of thesetraditional approaches has drawbacks if SOM runs with a small map size all centroids are located contiguously rather than attheir original distances according to the high-dimensional structure MS2Fs is not an efficient method because it does not takefeatures outside of the method into account and lastly PCA is a supervised method and loses the most valuable feature In thisstudy five novel hybrid approaches were proposed to eliminate these drawbacks by using the quantum genetic algorithm(QGA) method and four feature selection methods Pearsonrsquos correlation gain ratio information gain and relief methodsExperimental results demonstrate that for 14 datasets of different sizes the prediction accuracy of the proposed weightedclustering approaches is higher than the traditional K-means++ clustering approach Furthermore the proposed approachcombined with K-means++ and QGA shows the most efficient placements of the centroids on a two-dimensional map for allthe test datasets

1 Introduction

Human visual perception can be insufficient for the in-terpretation of a pattern within a multivariate (or highdimensional) structure causing errors at the decision-making stage In knowledge discovery processes thesame drawback is encountered in multivariate datasetsbecause of not being able to print the dataset to a visualinterface as a two-dimensional (2D) structure Further-more inefficient features in a multivariate dataset nega-tively impact the accuracy and running performance ofdata analysis tasks )erefore the notion of dimensionalreduction is of particular relevance in the preprocessingphase of data analysis Many algorithms and methods have

been proposed and developed for dimensional reduction[1] of which principal component analysis (PCA) is one ofthe most popular methods [2] Regardless of popularityneither PCA nor the other available dimensional reductionmethods are suitably efficient to independently visualizeall instances in the dataset because at the data prepara-tion stage before the usage of the data-mining algorithmPCA performs some feature selection tests separatelyfor different dimensions )ereafter the optimal featurenumber and the optimal model are determined withrespect to the variance values of these proposed models)e dataset is presented as a 2D map using PCA results ina low variance value Another drawback of PCA is thatthe most valuable feature is transformed to new values [2]

HindawiScientific ProgrammingVolume 2018 Article ID 9253295 23 pageshttpsdoiorg10115520189253295

As a result PCA is not typically successful at mappingthe dataset in 2D except in image representation andfacial recognition studies [3 4] Another difficulty isthat even if a dimensional reduction is applied the vi-sualization of all instances in a large dataset causes storageand performance problems To address this problemclustering algorithms can be used for data summarizationand the visualization methods applied afterwards K-Meansis the most-used clustering algorithm however K-meanssubmits only the high-dimensional centroids without anyvisualization and without any dimensional reductionoperation

Literature records three approaches to the visualizationof K-means )e first approach involves mapping for twoselected features from a multivariate dataset (MS2Fs)Curman et al have clustered the coordinate data inVancouver using K-means and presented them on a map[5] Cicoria et al have clustered ldquoTitanic Passenger Datardquousing K-means and printed the survival information thatwas the goal of the study to different 2D maps according tofeatures [6] an obtained binary image was clustered ontoa 2D interface using K-means for face detection in a studyby Hadi and Sumedang [7] and lastly Fang et al imple-mented software to visualize a dataset according to twoselected features [8] )e second approach is visualizationthat takes place after K-means clustering of the dataset isdimensionally reduced by PCA Nitsche has used this typeof visualization for document clustering [9] and Waghet al have used it for a molecular biological dataset [10])e third approach is the visualization of the datasetclustered by K-means in conjunction with an approach likeneighborhood method location in the self-organizing map(SOM) technique used by Raabe et al to implement a newapproach to show K-means clustering with a novel tech-nique locating the centroids on a 2D interface [11] SOM isa successful mapping and clustering algorithm neverthe-less it relies on the map-size parameter as the clusternumber and if it runs with a small map-size value allcentroids of the clusters are located contiguously not attheir original distances according to the high-dimensionalstructure

In our study new approaches are proposed to visu-alize the centroids of the clusters on a 2D map preservingthe original distances in the high-dimensional structure)ese different approaches are implemented as hybridalgorithms using K-means++ (an improved version ofK-means) SOM++ (an improved version of SOM)and the quantum genetic algorithm (QGA) QGA wasselected for use in our hybrid solutions because di-mensionality reduction and visualization problems formultivariate centroids (DRV-P-MC) are also an optimi-zation problem

)e contributions of this study are threefold firstclustering by K-means++ then mapping the centroids ontoa 2D interface using QGA and evaluating the success of thismethod A heuristic approach is proposed for DRV-P-MCand the aim is to avoid the drawback of locating the clusterscontiguously as in the traditional SOM++ Mapping thecentroids onto a 2D interface using SOM++ and evaluating

the success of this approach are performed to enablecomparison Second the usage of four major feature se-lection methods was mentioned in the paper by De Silva andLeong [12] specifically relief information gain gain ratioand correlation )e aim is to preserve the most valuablefeature and to evaluate it as the X axis Additionally the Yaxis is obtained by a weighted calculation using the co-efficient values returned from these feature selectionmethods )is provides an alternative to PCA for generatinga 2D dataset that avoids the PCA drawback of losing themost valuable feature )en clustering the datasets byK-means++ and mapping the centroids by these novelapproaches separately onto a 2D interface are performedMoreover mapping the centroids onto a 2D interface usingPCA and evaluating the success of this approach for com-parative purposes are performed )ird a versatile tool isimplemented with the capability to select the desired filealgorithm normalization type distance metric and sizeof the 2D map to calculate successes across six differ-ent metrics ldquosum of the square errorrdquo ldquoprecisionrdquo ldquorecallrdquoldquof-measurerdquo ldquoaccuracyrdquo and ldquodifference between multi-variate and 2D structuresrdquo )ese metrics are formulated indetail in Problem Definition

In literature generally MS2Fs has been the preferredmethod to manually determine the relations betweenfeatures Our tool is not only built on MS2Fs but alsocontains another algorithm that maps the centroids byusing information gain for comparison with our novelapproaches It assumes that the X axis is the most valuablefeature followed by the Y axis according to informationgain scores

)is paper details our study in seven sections DRV-P-MC is defined in detail in Section 2 related works providingguidance on DRV-P-MC including traditional algorithmshybrid approaches and the notion of dimensionality re-duction are submitted in Section 3 in Sections 4 and 5 thereorganized traditional approaches and the proposed al-gorithms are formulated and presented in detail the ex-perimental studies performed by using 14 datasets withdifferent characteristics and their accuracy results are givenin Section 6 and finally Section 7 presents conclusionsabout the proposed methods

2 Problem Definition

Considering that K-means++ is the major clustering algo-rithm used in this sort of problem the significance of patternvisualization in decision-making is clear However the vi-sualization requirement for the centroids of K-means++uncovers the need for DRV-P-MC since the centroidsreturned and the elements in them are presented by thetraditional K-means++ irrespective of any relation amongthe centroids )e aim in solving this problem is to map thecentroids onto a 2D interface by attaining the minimumdifference among the multivariate centroids and their ob-tained 2D projections

DRV-P-MC can be summarized as obtaining E from CTo detect whether the solution E of this problem is optimalor not ℓ must be measured as zero or a value close to zero

2 Scientific Programming

Two theorems and their proofs are described in this sectionwith an illustration for the measurement of ℓ

21 eorem 1 In measuring the distance between twomatrices like S and T performing the divisions of the valuesin S and T by the minimum distances in the matricesseparately avoids the revealed numeric difference owing tothe dimensional difference between E and C and thusa proportional balance between S and T is supplied as in (1)and (2)

S⟵ 1113944k

i11113944

k

ji+1Sij

1113944

f

a0Cia minusCja1113872 1113873

2

11139741113972

S⟵ 1113944k

i01113944

k

ji+1S[i j]

S[i j]

S[mi mj]

(1)

where Smimj is the minimum distance value in S

T⟵ 1113944k

i11113944

k

ji+1Tij

1113944

2

a0Eia minusEja1113872 1113873

2

11139741113972

T⟵ 1113944k

i01113944

k

ji+1T[i j]

T[i j]

T[mi mj]

(2)

where Tmimj is the minimum distance value in T

22 Proof 1 Focusing on the optimal M it can be observedthat the distance between the most similar centroids must belocated in the closest cells inM and the smallest values mustbe obtained for both S and T Moreover in this optimal Mthe other values in S and Tmust be obtained proportionallyto their smallest values

221 Illustration After clustering and placement opera-tions for k 4 f 3 r 6 and c 6 assume that C 0203 01 03 04 02 05 06 04 09 09 08 and E

0 0 0 3 2 2 5 5 and M is obtained as in thefollowing equation

M

mdash mdash mdash mdash mdash C3

mdash mdash mdash mdash mdash mdash


mdash mdash C2 mdash mdash mdash


C0 mdash mdash C1 mdash mdash

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

For M S and T matrices are obtained as the followingequations

S

0 01 09 11

0 0 03 09

0 0 0 05

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 17 28 112

0 0 22 53

0 0 0 42

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S

0 x 9x 11x

0 0 3x 9x

0 0 0 5x

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 y 16y 65y

0 0 12y 31y

0 0 0 24y

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArr

S

0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

where x and y isin R+ x is the minimum number in S and y isthe minimum number in T

23 eorem 2 To calculate the distance between twomatrices containing values that are balanced with each othertraditional matrix subtraction can be used )e subtractionoperations must be performed with the absolute values likethe approach in Manhattan distance to obtain a distancevalue greater than or equal to zero After the subtractionoperations a Z matrix is obtained and the sum of all valuesgives the difference between these matrices as follows

Z⟵ 1113944k

i11113944

k

ji+1Zij Sij minusTij

11138681113868111386811138681113868

11138681113868111386811138681113868

ℓ 1113944k

i11113944

k

ji+1Zij

(5)

Scientific Programming 3

24 Proof 2 To compare the two instances containing nu-meric values using machine learning normalization oper-ations must be performed to supply numeric balance amongthe features before the usage of any distance metric likeEuclidean distance or Manhattan distance )e first theo-rem essentially claims a normalization operation for S andTmatrices)e second theorem claims that with normalizedvalues the distance calculation can be performed by usingthe traditional subtraction operation

To illustrate both S and T have the smallest value as 1and normalized values )e closer the E is to the optimalsolution the smaller the values in the subtraction matrix Z)us ℓ can be obtained as a value close to zero In thisexample the value of ℓ is 74 + 55 + 18 + 59 + 26 232 asin the following equation

0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArrZ

0 0 74 55

0 0 18 59

0 0 0 26

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

3 Related Works

In our study some traditional machine learning algorithmswere utilized to implement hybrid approaches K-means++SOM++ PCA and QGA In this section these algorithmshybrid logic and dimensionality reduction are describedand reasons for preferences in our study are explained alongwith supporting literature

)e K-means++ algorithm is a successful clustering al-gorithm inspired by K-means that has been used in studiesacross broadly different domains For example Zhang andHepner have clustered a geographic area in a phenology study[13] Sharma et al have clustered satellite images in an as-tronomy study [14] Dzikrullah et al have clustered a pas-senger dataset in a study of public transportation systems [15]Nirmala andVeni have used K-means++ to obtain an efficienthybrid method in an optimization study [16] and lastlyWang et al have clustered a microcomputed tomographydataset in a medical study [17]

K-means++ is consistent in that it returns the samepattern at each run In addition the K-means++ algorithmsubmits related clusters for all instances and offers the ad-vantage of starting the cluster analysis with a good initial setof centers [18] Conversely it suffers the drawback of poorrunning performance in determining this initial set ofcenters and as to find good initial centroids it must performk passes over the data)erefore it was necessary to improveK-means++ for use of large datasets leading to the devel-opment of a more efficient parallel version of K-means++[19] Another study [20] addressed the problem by usinga sorted dataset which is claimed to decrease the runningtime)e literature also includes many studies on enhancingthe accuracy or the performance of K-means++ but none onvisualizing a 2D map for the clusters of this successful al-gorithm )erefore in this study a novel approach to vi-sualizing K-means++ clusters on a 2D map is detailed

SOM is both a clustering and a mapping algorithm usedas a visualization tool for exploratory data in different do-mains owing to its mapping ability [21] Each cluster in SOMis illustrated as a neuron and after the training process in anartificial neural network (ANN) structure each neuron hasX and Y values as a position on a map In addition allclusters in a SOM map are neighboring [22] Nanda et alused SOM for hydrological analysis [23] Chu et al usedSOM for their climate study [24] Voutilainen et al clustereda gerontological medical dataset using SOM [25] Kanzakiet al used SOM in their radiation study to analyze the liverdamage from radon X-rays or alcohol treatments in mice[26] and Tsai et al have clustered a dataset about water andfish species in an ecohydrological environment study [27]

Although SOM produces a map where each neuronrepresents a cluster and all clusters are neighboring [21 22] themap does not position the returned centroids adjacent to eachother as 1-unit distances in fact these clusters must be locatedon farther cells of the map In this study a novel approach tovisualize SOM mappings in 2D retaining practically originaldistances among clusters is detailed using the fast versionof SOM SOM++ In SOM++ the initialization step ofK-means++ is used to find the initial centroids of SOM [28]

PCA is another popular machine learning method forfeature selection and dimensionality reduction Owing to itsversatility this algorithm is also used across different domainsViani et al used PCA to analyze channel state information(CSI) for wireless detection of passive targets [29] Tiwari et alanalyzed solar-based organic ranking cycles for optimizationusing PCA [30] Hamill et al used PCA to sort multivariatechemistry datasets [31] Ishiyama et al analyzed a cytomega-lovirus dataset in a medical study [32] and Halai et al usedPCA to analyze a neuropsychological model in a psychologystudy [33] PCA is used to eliminate some features for mul-tivariate datasets before machine learning analysis as thedataset may be large and contain some features that wouldmake analysis efficient [34])ese unnecessary featuresmay beidentified by PCA for subsequent removal resulting in a newdataset with new values and fewer features [2] Essentially thisalgorithm is not a clustering algorithm however PCA isrelevant owing to its dimensionality reduction capacity this isutilized in our study to obtain a new 2D dataset which is thenclustered using the traditional K-means++ approach

QGA is a heuristic optimization algorithm It refers to thesmallest unit storing information in a quantum computer asa quantum bit (qubit) A qubit may store a value in betweenthe binary values of ldquo1rdquo or ldquo0rdquo significantly decreasing run-ning time in the determination of an optimized result [35 36])is contemporary optimization algorithm is in wide-spreaduse Silveira et al used QGA to implement a novel approachfor ordering optimization problems [37] Chen et al usedQGA for a path planning problem [38] Guan and Linimplemented a system to obtain a structural optimal designfor ships using QGA [39] Ning et al used QGA to solve a ldquojobshop scheduling problemrdquo in their study [40] and Konar et alimplemented a novel QGA as a hybrid quantum-inspiredgenetic algorithm to solve the problem of scheduling real-timetasks in multiprocessor systems [41] DRV-P-MC the focus ofinterest in our study is an optimization problem as well so


this algorithmrsquos efficient run-time performance is employed todetermine suitable cluster positions on a 2D map

Hybrid approaches combine efficient parts of certain al-gorithms into new wholes enhancing the accuracy and theefficiency of these algorithms or producing novel algorithmsIn literature there are many successful hybrid data-miningapproaches Kumar et al implemented a highly accurateoptimization algorithm from the combination of a geneticalgorithm with fuzzy logic and ANN [42] Singhal and Ashrafimplemented a high-performance classification algorithmfrom the combination of a decision tree and a genetic algo-rithm [43] Hassan and Verma collected successful high-accuracy hybrid data-mining applications for the medicaldomain in their study [44] )amilselvan and Sathiaseelanreviewed hybrid data-mining algorithms for image classifi-cation [45] Athiyaman et al implemented a high-accuracyapproach combination of association rule mining algorithmsand clustering algorithms for meteorological datasets [46]Sahay et al proposed a high-performance hybrid data-miningapproach combining apriori and K-means algorithms forcloud computing [47] Yu et al obtained a novel solutionselection strategy using hybrid clustering algorithms [48] Sitekand Wikarek implemented a hybrid framework for solvingoptimization problems and constraint satisfaction by usingconstraint logic programming constraint programming andmathematical programming [49] Abdel-Maksoud et al pro-posed a hybrid clustering technique combining K-means andfuzzy C-means algorithms to detect brain tumours with highaccuracy and performance [50] Zhu et al implementeda novel high-performance hybrid approach containing hier-archical clustering algorithms for the structure of wirelessnetworks [51] Rahman and Islam combined K-means anda genetic algorithm to obtain a novel high-performance geneticalgorithm [52] and Jagtap proposed a high-accuracy tech-nique to diagnose heart disease by combining Naıve BayesMultilayer Perceptron C45 as a decision tree algorithm andlinear regression [53] What we can infer from a detailedexamination of these studies is that K-means and geneticalgorithms and their variants can be adapted to other algo-rithms to implement a hybrid approach successfully More-over the combination of K-means and genetic algorithmscreates an extremely efficient and highly accurate algorithm

In data analysis unnecessary features cause two mainproblems in performance and accuracy If a dataset is large orhas insignificant features a downscaling process should beperformed by a dimensionality reduction operation to enableefficient use of the analysis algorithms In literature manytechniques related to dimensionality reduction are presentedFor example Dash et al claimed that using PCA for di-mensionality reduction causes a drawback in understandingthe dataset owing to the creation of new features with newvalues Furthermore they posit that the most effective at-tributes are damaged )erefore they presented a novel ap-proach based on an entropy measure for dimensionalityreduction [54] Bingham and Mannila used a random pro-jection (RP) method instead of PCA for the dimensionalityreduction of image and text datasets singular value de-composition (SVD) latent semantic indexing (LSI) anddiscrete cosine transform claiming that RP offers simpler

calculation than the other methods and has low error rates[55] Goh and Vidal have used k-nearest neighbor andK-means to obtain a novel method for clustering and di-mensionality reduction on Riemannian manifolds [56] Na-poleon and Pavalakodi implemented a new technique usingPCA and K-means for dimensionality reduction on high-dimensional datasets [57] Samudrala et al implementeda parallel framework to reduce the dimensions of large-scaledatasets by using PCA [58] Cunningham and Byron usedPCA factor analysis (FA) Gaussian process factor analysislatent linear dynamical systems and latent nonlinear dy-namical systems for the dimensional reduction of humanneuronal data [59] and Demarchi et al reduced the di-mensions of the APEX (airborne prism experiment) datasetusing the auto-associative neural network approach and theBandClust algorithm [60] Boutsidis et al implemented twodifferent dimensional reduction approaches for K-meansclustering the first based on RP and the second based on SVD[61] Azar and Hassanien proposed a neurofuzzy classifiermethod based on ANN and fuzzy logic for the dimensionalreduction of medical big datasets [62] Cohen et al imple-mented a method using RP and SVD for the dimensionalreduction in K-means clustering [63] Cunningham andGhahramani discussed PCA FA linear regression Fisherrsquoslinear discriminant analysis linear multidimensional scalingcanonical correlations analysis slow feature analysisundercomplete independent component analysis sufficientdimensionality reduction distance metric learning andmaximum autocorrelation factors in their survey article andobserved that in particular PCA was used and evaluated inmany studies as a highly accurate analysis [1] and Zhao et alused 2D-PCA and 2D locality preserving projection for the2D dimensionality reduction in their study [64] Sharifzadehet al improved a PCA method as sparse supervised principalcomponent analysis (SSPCA) to adapt PCA for di-mensionality reduction of supervised datasets claiming thatthe addition of the target attribute made the feature selectionand dimensional reduction operations more successful [65]

)ese studies show PCA to be the most-used method fordimensionality reduction despite reported disadvantagesincluding the creation of new features which may hamperthe understanding of the dataset changing the values in themost important and efficient features and complex calcu-lation and low performance for big datasets

In other sample clustering studies Yu et al proposed somedistribution-based distance functions used to measure thesimilarity between two sets of Gaussian distributions in theirstudy and distribution-based cluster structure selection Ad-ditionally they implemented a framework to determine theunified cluster structure from multiple cluster structures in alldata used in their study [66] In another study by Yu andWong a quantization driven clustering approach was designedto obtain classes for many instances Moreover they proposedtwo different methods to improve the performance of theirapproach the shrinking process and the hierarchical structureprocess [67] A study by Wang et al proposed a local gravi-tationmodel and implemented two novel measures to discovermore information among instances a local gravitation clus-tering algorithm for clustering and evaluating the effectiveness


of the model and communication with local agents to attainsatisfactory clustering patterns using only one parameter [68]Yu et al designed a framework known as the double affinitypropagation driven cluster for clustering on noisy instancesand integrated multiple distance functions to avoid the noiseinvolved with using a single distance function [69]

4 The Reorganized Traditional Approaches

)is paper assumes two ways of K-means++ clustering thetraditional usage and a weighted usage After normalizationtechniques are used to balance the dataset features andnormalize the dataset traditional K-means++ clustering isperformed )is has a preprocess step to discover the initialcentroids for the standard K-means After the initializationof the centroids K-means clustering runs using the initialcentroid values K-means++ is expressed as Algorithm 1

In this study three reorganized traditional mappingapproaches were implemented to visualize the centroids ofK-means++ returned by Algorithm 1 mapping by SOM++by PCA and according to the best two features determinedby information gain )ese three algorithms are the ap-proaches that were implemented previously in the literatureTo compare these approaches with the other five proposedalgorithms under the same conditions the former threealgorithms are appropriately reorganized and implementedin this study and formulated in the following manner

41 K-Means++ and Mapping by SOM++ (KS) )e best-knownmapping algorithm is SOM and in this study SOM++was implemented as the improved version of SOM providingthe centroids generated by K-means++ as the initial weightsfor SOM to solve DRV-P-MC In this approach the traditionalK-means++ clustering runs first and then returns the cen-troids to bemapped)e SOM++ algorithm trains the weightsof the neurons on its map by using the Gaussian function asthe update function as in (7) After a specific number ofiterations the weight values reach a trained-value stateaccording to the centroids To map the centroids the winningneurons with the nearest distance for all centroids separatelyare calculated and the weight values of these winning neuronsare converted to the multivariate values of the related cen-troids as in (8) Finally M containing the pure multivariatevalues of the centroids as the weight values is returned

Let us assume that C is obtained by Algorithm 1 θ is theclosest cell in M for each element in C d is the minimumdistance between the current instances and Cc and θ and foreach neighbor of θ is computed as in the following equation

h expminusd2

2σ21113888 1113889

ω⟵ 1113944

f

i1ωi ωi + hlowast ηlowast Cci minusωi1113872 1113873

(7)

where α is the neighborhood width parameter η is thelearning rate parameter h is the neighbourhood functionand ω is the set of weight values in each cell in M

1113944

k

i11113944

f

j1WC(i j) C(i j) (8)

whereWC is the set of the winner cells for each element in C

42 K-Means++ and Mapping by PCA (KP) )e next ap-proach implemented in this study to map the centroids ofK-means++ incorporates evaluated PCA In this approachonce again the traditional K-means++ clustering runs firstand then returns the centroids to be mapped )e PCAalgorithm is a supervised learning algorithm therefore thedataset cannot be used by PCA without a target attribute Inthe next step to enable the use of PCA another dataset wascreated with all the features in the original version and anextra attribute to function as the target attribute )is newattribute is filled for each instance according to the obtainedcentroids by calculating the distances among the instancesand these centroids Finally PCA calculates two componentsby using this new dataset and considering the target attri-bute and a 2D structure is attained to map the centroids onM )is approach is formulated in the following expressions

Let us assume that C is obtained by Algorithm 1 Ωcomputed by C is the set of the clusters of the instances in Iand Iprime is a new set of the instances with a new attributecreated as the target by filling it with Ω

Let us assume that for matrix L Lx and Ly are the resultmatrices as shown in the following equations respectively

Lx LlowastLT

Ly LT lowastL(9)

An alternative way of computing Ly is as in the followingequation

EVx LlowastEVy lowastEL12

Ly EVy lowastELlowastEVyT

L⟵1113944n

i1Iiprime minus micro( 1113857

Ly⟵1113944

n

i1L

T lowastL

(10)

where EL is the same positive eigenvalue of Lx and Ly EVx isthe eigenvectors of Lx EVy is the eigenvectors of Ly and μ isthe mean vector of the instances in I

IPCA is the 2D dataset computed by PCA as in thefollowing equation

IPCA UT2 lowast L (11)

where U2 [u1 u2] is the set of the two first components forthe 2D structure

In the next expressions the aim is to obtain 2D centroidsequivalent to their multivariate structure)erefore firstly themeans of two new features of the instances in the same clusterare calculated for each cluster and each feature separately Letus assume that Ω is valid for the 2D instances in IPCA andCPCA is calculated as 2D centroids as in the following equation


CPCA⟵ 1113944n

i11113944

2

j1CPCA(Ωi j) CPCA(Ωi j) + IPCA(i j)

CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)

Moreover the map has the number of its column between0 and c and the number of its row between 0 and r and PCAreturns two features as decimal numbers )erefore secondlyan operation for normalization must be implemented to placethese centroids on the map having ctimes r dimensions )usthese decimal numbers shift their integer equivalents by usingthe min-max normalization technique)e centroids can thenbe placed on the ctimes rmap FinallyM in which the centroids inE are mapped is obtained as in the following equation

E⟵ 1113944k

i11113944

2

j1E(i j)

CPCA(i j)minusminj1113960 1113961lowast c

maxj minusminj1113872 1113873 (13)

where NC1 is the set of the first column values in CPCA NC2is the set of the second column values in CPCA minNC1 is theminimum value in NC1 maxNC1 is the maximum value inNC1 minNC2 is the minimum value in NC2 and maxNC2 isthe maximum value in NC2

43 K-Means++ and Mapping according to the Best TwoFeaturesDetermined by InformationGain (B2FM) )e third

approach mapping the centroids by information gainaccording to the best two features is formulated in thefollowing expressions Firstly K-means++ runs and thecentroids are obtained )e aim in this approach is toevaluate only the most valuable features for mappingtherefore the information gain method is used computed todetermine the root feature by decision tree algorithms Twofeatures having the highest information gain scores areconsidered as the X axis and the Y axis of the map )einformation gain method is a supervised learning algorithmand it needs a target attribute like PCA For this reason inthe second step of this approach a new dataset with thetarget attribute is created by evaluation of the distancesbetween all centroids and all instances )us the infor-mation gain method can use this new dataset and obtain thescores for each feature At the end of this approach thevalues in these features in the centroids shift their integerequivalents between 0 and c and between 0 and r by using themin-max normalization technique Finally the centroids areplaced on the c x r map

Let us assume that C is obtained by Algorithm 1 Ωcomputed by C is the set of the clusters of the instances in IIprime is a new set of the instances using a new attribute as thetarget (by filling it with Ω) IG is obtained by informationgain feature selection method with Iprime expressed in (23) fcis the highest ranked feature in IG and FC is the set ofthe values in the fcth feature in C as in (14) sc is thesecond highest ranked feature in IG and SC is the set of thevalues in the scth feature in C as shown in the followingequations

Input Number of the centroids kOutput Set of the final centroids CBeginv≔Randomly selected element in IV Set of the initial centroidsY Set of the distances between v and the closest element to v in V

y Length of YAdd v into V

Add the distance between v and the closest element to v in V into YRepeatFor i 1 yU≔U+Y2

i

u≔A random real number between 0 and Up≔ 2RepeatU prime≔ 0For i 1 pminus 1U prime≔Uprime+Y2

i

p≔ p+ 1Until Uge ugtUprimeAdd Ipminus1 into V

Add the distance between Ipminus1 and the closest element to Ipminus1 in V into YUntil V has k centroidsRun the standard K-means with V

Return C returned from the standard K-meansEnd

ALGORITHM 1 K-means++


FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)

Finally M where the centroids in E are mapped isobtained as in the following equation

E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)

where minfc is the minimum value in FC maxfc is themaximum value in FC minfc is the minimum value in FCand maxfc is the maximum value in FC

5 The Proposed Approaches

)e five approaches proposed in this study are K-means++using QGA mapping and weighted K-Means++ usingmapping by Pearsonrsquos correlation gain ratio informationgain and relief feature selection methods )ese five ap-proaches are implemented in the study and formulated inthe following manner

51 K-Means++ and Mapping by QGA (KQ) )e firstproposed approach in our study uses QGA to map thecentroids of K-means++ Solving DRV-P-MC can be con-sidered as a heuristic approach because k numbers ofcentroids are tried for placement on a cminus r dimensional mapmeaning that there are (clowast r)lowast (clowast rminus 1)lowast (clowast rminus 2)lowast(clowast rminus 3)lowast middot middot middot lowast (clowast rminus k+ 1) probabilities )erefore thisproblem resembles the well-known ldquoTravelling Salesmenrdquo

optimization problem Given the large number of proba-bilities the fittest map pattern can be attained by an opti-mization algorithm In this study QGA the improvedversion of genetic algorithms is used In QGA the chro-mosomes represent the 2D map matrixM and the genes inthe chromosomes represent the cells in M Moreover eachgene in QGA has a substructure named ldquoqubitrdquo with 2components α and β α represents the probability of theexistence of the evaluated centroid for the current cell and βrepresents the probability of the absence of the evaluatedcentroid for the current cell )is multicontrol mechanismhelps QGA achieve high performance for a large search space

)e first operation is initialization of each qubit (α β)with 1

2

radicin genes identically In the second step the aim is

to check the probabilities of placing the centroids into thecells separately and to obtain the map in which all centroidsare placed as a candidate map )ese operations are per-formed by the makeOperator() method using a randomi-zation technique containing three random numbers Two ofthem specify the column and the row and the third de-termines the cell of the current centroid [36] )e fitnessfunction in this approach calculates the differences betweenE and C which is described in detail in Problem Definition)erefore the value returned by the fitness function isideally approximately zero QGA has a generation number asa parameter and if this generation number of iterationscompletes the best individual with the lowest fitnessfunction value is assumed as the fittest map If not QGAneeds another assistant method called updateOperator()

Let us assume that rc is the random column numberattained by random [0 c) rr is the random row numberattained by random [0 r) rn is the random number attainedby random [0 1) and Mprime returned by makeOperator() iscomputed as in the following equation

Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0

obtain new rc and rr numbers decrease it

and for the current i continue to control otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

)e updateOperator() method updates all qubits in theindividuals in the population according to the best indi-vidual to approximate its angularity using a trigonometricapproach Firstly this method needs a subfunction called thesign function sf to obtain a score between 0 and 1 accordingto the distance between the fitness function values of the bestindividual and the individual being evaluated After that thissf value is evaluated in a lookup table containing a conditionlist to determine the angle [36]

Let us assume that ff is the value of the fitnessfunction BS is the best individual in the previous

population according to ff(BS) CS is the current indi-vidual in the current population according to ff(CS) bvis a gene value obtained by makeOperator() for BS cv isa gene value obtained by makeOperator() for CS αprime and βprimeare the previous values of α and β for CS sf is the signfunction and its value is extracted according to theconditions list as in (18) ∆z is the orientation of rotationangle to update the qubit values as in (19) and α and β areupdated for each gene in each chromosome in QM as in(20) Finally QM with new qubit values is returned byupdateOperator()


sf αprime βprime( 1113857

+1 αprime lowast βprime lowast [ff(BS)minus ff(CS)]lt 0

minus1 αprime lowast βprime lowast [ff(BS)minus ff(CS)]gt 0

plusmn1 αprime 0 and (bvminus cv)lowast [ff(BS)minus ff(CS)]lt 0

plusmn1 βprime 0 and (bvminus cv)lowast [ff(BS)minus ff(CS)]gt 0

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz

0025π ff(BS)ge ff(CS) and cv 1 and αprime 0 and βprime minus1 and sf αprime βprime( 1113857 plusmn1

0005π ff(BS)lt ff(CS) and cv 1 and bv 1 and αprime 0 and βprime minus1 and sf αprime βprime( 1113857 +1

001π ff(BS)lt ff(CS) and cv 1 and bv 0 and αprime plusmn1 and βprime +1 and sf αprime βprime( 1113857 +1

005π ff(BS)ge ff(CS) and cv 0 and bv 1 and αprime plusmn1 and βprime +1 and sf αprime βprime( 1113857 minus1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ

1113890 1113891 cos(Δz) minussin(Δz)

sin(Δz) cos(Δz)1113890 1113891lowast

αprime

βprime1113890 1113891 (20)

)e calling of the makeOperator() and updateOperator()methods continues until the generation number is reachedAfter completing the iterations the fittest map is returned bythe algorithm as given in Algorithm 2

52 Weighted K-Means++ and Mapping by Feature SelectionMethods Literature indicates that when the supervisedanalysis of a dataset involves many features insignificantfeatures can negatively affect the results of the analysis)erefore feature selection methods are used to eliminatesome features according to their degrees of influence overthe class attribute In the next proposed approaches allfeatures can be considered as being determined by the X andY axes without any elimination and with losing the mostvaluable feature unlike in PCA even if 2D mapping needsonly two dimensions Indeed the aims of these approachesare to keep the most valuable feature and to assume it to bethe X axis and in addition to evaluate other all features ina weighted manner as the Y axis As a result these ap-proaches use four feature selectionmethods to determine theefficiencies of the features separately Pearsonrsquos correlationgain ratio information gain and relief In these proposedapproaches the centroids obtained by the traditionalK-means++ clustering are not evaluated in their pure form

but are converted to their weighted values according to thescores returned by the feature selection methods

In Algorithm 3 the weighted K-means++ clustering isexpressed Firstly the values of the features in all instances aremultiplied by the weight values and a new set of the instancesis attained Finally these transformed instances are used by thetraditional K-means++ clustering and the more qualifiedcentroids are returned by this method )e efficiency and thesuccess of theweighted clustering approach are detailed in [70]

521 Weighted K-Means++ and Mapping by PearsonrsquosCorrelation (WMC) In all approaches in this study un-supervised analyses are the focus therefore to use featureselection methods which are supervised learning algo-rithms firstly the traditional K-means++ clustering isimplemented and a new dataset with the clusters returned iscreated )e correlation coefficient values are formulated inthe following expressions for all features can be calculatedwith this dataset thus these values are evaluated by weightedK-means++ clustering and the centroids that are returnedcan be placed on the 2D map

Let us assume that CC is the set of the correlation co-efficient values between all features and the target attribute asin the following equation

CC⟵ 1113944

f

i1

nlowast1113936nj1 Iij lowast tj1113872 1113873minus1113936

nj1Iij lowast1113936

nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi

mincc mincc gt 0gt 0

maxccCCi

maxcc lt 0

CCi minusminccmaxcc minusmincc

lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)


where mincc is the minimum value in CC and maxcc is themaximum value in CC

Algorithm 4 formulated to preserve the most valuablefeature determined by Pearsonrsquos correlation gives a detailedpresentation of the weighted mapping of the centroids onthe 2D map In this method the values in the most valuablefeature and the weighted averaged values in the other fea-tures in the centroids shift their integer equivalents between0 and c and between 0 and r by using the min-max nor-malization technique Finally M is returned

522 Weighted K-Means++ and Mapping by Gain Ratio(WMG) In Algorithm 5 the weighted K-means++ usinggain ratio feature selection and weighted mapping by

preserving the most valuable feature are formulated )esame operations as in Algorithm 4 are implemented in thisproposed approach except the calculation of the weightvalues )e weight values are obtained by the gain ratiofeature selection method in this approach In additionowing to the different weight values from the correlationcoefficient values centroids with different multivariatefeature values are computed in this mapping method

Let us assume that Bi represents the ith category in tP(Bi) is the probability of the ith category Gj represents thejth category in a feature P(Gi) is the probability of the jthcategory P(Bi|Gj) is the conditional probability of the ithcategory given that term Gj appeared and GR is the set ofthe gain ratio values between all features and the targetattribute as shown in the following equation

Input Number of the centroids k iteration number (it)Output Map with C placed MBeginQM Population containing individual qubit matricesPM Population containing individual probability matricesqm Length of QMpm Length of PMtr≔ 0C Set of the centroids returned by Algorithm 1RepeatFor a 1 qmFor i 1 cFor j 1 rQM middot αaij ≔ 1radic2QM middot βaij ≔ 1radic2

For a 1 pmmakeOperator(it PMa)

QM updateOperator(QM)tr≔ tr + 1Until tr itReturn M having the fittest valueEnd

ALGORITHM 2 K-means++ and mapping by QGA

Input Set of the weight values for all features (W)Output Set of the final centroids CBeginFor i 1 n

For j 1 fIij Iij lowastWj

Call Algorithm 1 for I to cluster with K-means++Return C returned by Algorithm 1End

ALGORITHM 3 )e weighted K-means++ clustering


GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

minus1113936gj1P Gi( 1113857 log P Gi( 1113857

GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0

GRi minusmingrmaxgr minusmingr

lowast 09 + 01 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where mingr is the minimum value in GR and maxgr is themaximum value in GR

523 Weighted K-Means++ and Mapping by InformationGain (WMI) In Algorithm 5 the weighted K-means++using information gain feature selection and weightedmapping by preserving the most valuable feature isformulated )e same operations are implemented as inthe previous approach except the calculation of theweight values )e weight values are obtained by the in-formation gain feature selection method in this approach

Furthermore because of the different weight values fromgain ratio values centroids with different multivariatefeature values are computed in this mapping method(Algorithm 6)

Let us assume that Bi represents the ith category in t andP(Bi) is the probability of the ith category Gj represents thejth category in a feature and (Gj) is the probability of the jthcategory P(Bi|Gj) is the conditional probability of the ithcategory given that term Gj appeared and IG is the set of theinformation gain values between all features and the targetattribute

Input Number of the centroids kOutput Map with C placed MBeginCprime Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 1Ω Set of the clusters of the instances in I computed by CprimeIprime Set of the instances with a new attribute as the target by filling it with ΩCC Set of the weight values obtained by Pearsonrsquos correlation feature selection methodC Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 3fc )e highest ranked feature in CCFC Set of the values in the fcth feature in Cwc Sum of the scores in the features except the fcth featureWC Set of the average of the values in the other featuresFor i 1 kFCi Cifc

For i 1 kFor j 1 fIf j is not equal to fcWCi WCi+Cij

For i 1 fIf j is not equal to fcwcwc+CCi

For i 1 kWCi WCiwc

minfc )e minimum value is in FCmaxfc )e maximum value is in FCminwc )e minimum value is in WCmaxwc )e maximum value is in WCFor i 1 kFor j 1 fEij [Cijminusminj]lowast c(maxjminusminj)

Return M where the centroids in E are mappedEnd

ALGORITHM 4 )e weighted K-means++ and mapping by Pearsonrsquos correlation


Input Number of the centroids kOutput Map with C placed MBeginCprime Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 1Ω Set of the clusters of the instances in I computed by CprimeIprime Set of the instances with a new attribute as the target by filling it with ΩGC Set of the weight values obtained by gain ratio feature selection methodC Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 3fc )e highest ranked feature in GRFC Set of the values in the fcth feature in Cwc Sum of the scores in the features except the fcth featureWC Set of the average of the values in the other featuresFor i 1 kFCi Cifc

For i 1 kFor j 1 fIf j is not equal to fcWCi WCi+Ci j

For i 1 fIf j is not equal to fcwcwc+GRi

For i 1 kWCi WCiwc



ALGORITHM 5 )e weighted K-means++ and mapping by gain ratio

Input Number of the centroids kOutput Map with C placed MBeginCprime Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 1Ω Set of the clusters of the instances in I computed by CprimeIprime Set of the instances with a new attribute as the target by filling it with ΩIG Set of the weight values obtained by information gain feature selection methodC Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 3fc )e highest ranked feature in IGFC Set of the values in the fcth feature in Cwc Sum of the scores in the features except the fcth featureWC Set of the average of the values in the other featuresFor i 1 kFCi Cifc

For i 1 kFor j 1 fIf j is not equal to fcWCiWCi+Cij

For i 1 fIf j is not equal to fcwcwc+ IGi

For i 1 kWCi WCiwc

minfc )e minimum value is in FC

ALGORITHM 6 Continued


IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0

IGi minusminIGmaxIG minusminIG

lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where minIG is the minimum value in IG and maxIG is themaximum value in IG

524 Weighted K-Means++ and Mapping by Relief (WMR)In Algorithm 7 the weighted K-means++ using relief featureselection and weighted mapping by preserving the mostvaluable feature is formulated )e same operations as in theprevious approach are implemented except the calculationof the weight values )e weight values are obtained by therelief feature selection method in this approach Moreoverbecause of the different weight values from information gainvalues centroids with different multivariate feature valuesare computed in this mapping method Let us assume that Ris a random instance fromD Ih is the closest instance to R inthe same class where R exists Im is the closest instance to Rin another class where R does not exist and RF is the set ofthe relief feature values between all features and the targetattribute in the following equation

RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f

j1RFj RFj minus Ihj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0

RFi minusminrfmaxrf minusminrf

lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where minrf is the minimum value in RF and maxrf is themaximum value in RF

6 Experimental Results and Discussions

Accuracy consistency and compatibility tests are imple-mented in the study)e accuracy tests involvemeasurementsof precision recall f-measure and accuracy Consistency testsinvolve calculation of the sum of squared error (SSE)Compatibility tests consider the difference between multi-variate and 2D structures (DBM2) )is section details theimplementation of the tests on the proposed and traditionalalgorithms on 14 datasets of various sizes and comparison ofthe test results For the measurement of the relevant metricsfor these processes a versatile tool whose interface is shownin Figure 1 was developed as a desktop application on VisualStudio 2017 using the WEKA data-mining software packagefor background operations [71] To implement this toolprogramming aspects included the development of pseudo-code (fragments were given in Section 4 and Section 5) withconsideration of mainly object-oriented programming con-cepts leading to the placement of each algorithm as a classstructure within the application Finally C was selected asa programming language in which sorted lists were imple-mented as data structures

)e test tool was developed with the capability to selectthe desired file algorithm size of the 2Dmap normalizationtype and distance metric Aside from the algorithms theoptions are 2D map sizes from 2times 2 to 20times 20 Euclidean(25) Manhattan (26) or Chebyshev (27) distance metricsand four normalization types specifically min-max (28)z-score (29) decimal scaling (30) and division by maximumvalue methods (31)

dij

1113944

f

a1Iia minus Ija1113872 1113873

2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)

where dij is the distance between the ith and jth instances

maxfc )e maximum value is in FCminwc )e minimum value is in WCmaxwc )e maximum value is in WCFor i 1 kFor j 1 fEij [Cijminusminj]lowast c(maxjminusminj)


ALGORITHM 6 )e weighted K-means++ and mapping by information gain


Input Number of the centroids kOutput Map with C placed MBeginCprime Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 1Ω Set of the clusters of the instances in I computed by CprimeIprime Set of the instances with a new attribute as the target by filling it with ΩRF Set of the weight values obtained by information gain feature selection methodC Set of the centroids obtained by the traditional K-means++ clustering in Algorithm 3fc )e highest ranked feature in RFFC Set of the values in the fcth feature in Cwc Sum of the scores in the features except the fcth featureWC Set of the average of the values in the other featuresFor i 1 k

FCi CifcFor i 1 k

For j 1 fIf j is not equal to fcWCi WCi+Cij

For i 1 fIf j is not equal to fcwcwc+RFi

For i 1 kWCi WCiwc

minfc )e minimum value is in FCmaxfc )e maximum value is in FCminwc )e minimum value is in WCmaxwc )e maximum value is in WCFor i 1 k

For j 1 fEij [Cij -minj] lowast c(maxj -minj)


ALGORITHM 7 )e weighted K-means++ and mapping by relief

Figure 1 An interface of the tool after clustering


NV FVminus FVmin

FVmax minus FVminlowast NVmax minusNVmin( 1113857 + NVmin (28)

NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)

where FV is the current value NV is the normalized value ofFV FVmin is the minimum value in the current feature (F)FVmax is the maximum value in F NVmax is the new max-imum value in F NVmin is the new minimum value in F micro isthe mean of the values in F b is the standard deviation valuein F and τ is lowest integer value such that max (|NV|)lt 1

61 Dataset Description )e datasets used in our study arelisted in Table 1 together with their characteristics )ey areopen-source datasets from the UCI machine learning re-pository and are the same datasets used in previous studiesin the literature containing supervised analyses because oftheir class attributes

62 Validation Metrics To verify the accuracy of the algo-rithms blind versions of these datasets were created in theexperimental studies and the results of the precision (Pr) recall(Re) f-measure (Fm) and accuracy (Ac) tests were generated bymeans of matches between the desired and proposed classes)e Pr Re Fm andAc test results in this section were calculatedusing min-max normalization between 0 and 1 and using theEuclidean distance metric owing to this being generally usedfor K-means++ clustering in literature In addition the clus-tering operations were performed according to the originalclass numbers of the datasets in order to be able to compare thedesired classes with the proposed clusters

In Table 2 the Ac Fm Re and Pr values which werecomputed by the standard K-means++ clustering and theweighted K-means++ clustering with Pearsonrsquos correlationgain ratio information gain and relief features selectionmethods are given )ey are presented separately as per-centages for 14 datasets in Table 2 )e Ac Fm Re and Prvalues can be computed as follows respectively

Ac Tp + Tn

Tp + Tn + Fp + Fn (32)

Fm 2lowastPr lowastRe

Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)

where Tp is the true-positive value Tn is the true-negativevalue Fp is the false-positive value and Fn is the false-negativevalue

To verify the consistency of the centroids the measure-ments for SSE analyses were implemented in this experi-mental study )e SSE values for 14 datasets using EuclideanManhattan and Chebyshev distance metrics to see the var-iances for each metric were computed separately SSE iscomputed as in the following equation

SSE 1113944k

i11113944

f

j1Cij minus IPrimeij1113872 1113873

2 (36)

where Iiprime is the set of the closest instances in I to Ci

Finally for compatibility testing the differences betweenmultivariate and 2D structures of the centroids as ℓ weremeasured as detailed in Problem Definition )e ℓ value isexpected to be zero or approximately zero Resultant ℓvalues were evaluated separately in terms of the traditional(KS KP and B2FM) and the proposed (KQ WMC WMGWMI andWMR) approaches )e results shown in Figure 2were obtained on 10 datasets having more than 2 classesbecause it was observed that all approaches could map the 2centroids of the other 4 datasets (Heart Statlog Johns HopkinsUniversity Ionosphere Database Pima Indians DiabetesDatabase and Wisconsin breast cancer) on the furthest edgeson the map and consequently the ℓ values were computed as0 that is the most optimal value that can be obtained

63 Experimental Results )e results of accuracy tests showgenerally that information gain is the most successfulclustering approach as it has higher Ac values for 12 datasetsthan is standard in comparison with the other approachesGain ratio has higher accuracy values than standard for 9datasets correlation has higher accuracy for 7 datasets andrelief has higher accuracy for 5 datasets Considering Fmvalues information gain has higher values than standard for10 datasets gain ratio has higher values for 8 datasetscorrelation has higher values for 6 datasets and relief hashigher values for 4 datasets With respect to Re informationgain has higher values then standard for 10 datasets gainratio has higher values for 8 datasets correlation has highervalues for 6 datasets and relief has higher values 5 fordatasets Finally regarding Pr values info gain has higherpercentage values for 9 datasets than standard gain ratio hashigher values for 8 datasets correlation has higher values for6 datasets and relief has higher values for 3 datasets )eseresults support the conclusion that weighted K-means++clustering with feature selection methods particularly in-formation gain provides more accurate centroids thantraditional K-means++ clustering

)e values in Table 3 were obtained by calculating themeans of the Ac Fm Re and Pr values in Table 2 for 14datasets )e results in this table demonstrate that theweighted clustering approaches help to increase the accuracyof the clustering algorithms In addition information gain isshown to be the most efficient feature selection method forclustering operations as it has the highest values for the


Table 1 )e datasets with their types existence of missing values and sizes

Datasets Type Instances Features Classes Missing valuesBlocks Classification Numeric 5473 10 5 NoCardiac Arrhythmia Database Numeric 452 278 16 YesColumn 3C Weka Numeric 310 6 3 NoGlass Identification Database Numeric 214 9 7 NoHeart Statlog Numeric 270 13 2 NoIris Plants Database Numeric 150 4 3 NoJohns Hopkins University Ionosphere Database Numeric 351 34 2 NoOptical Recognition of Handwritten Digits Numeric 5620 64 10 NoPima Indians Diabetes Database Numeric 768 8 2 NoProtein Localization Sites Numeric 336 7 8 NoVowel Context Data Numeric 990 11 11 NoWaveform Database Generator Numeric 5000 40 3 NoWine Recognition Data Numeric 178 13 3 NoWisconsin Breast Cancer Numeric 699 9 2 Yes

Table 2 )e Ac Fm Re and Pr values () calculated separately for 14 datasets and the standard and the weighted K-means++ clusteringapproaches

Datasets Standard Correlation Gain ratio Info gain Relief

Blocks Classification

Ac 9481 9513lowast 9493lowast 9512lowast 9513lowastFm 9537 9559lowast 9545lowast 9558lowast 9559lowastRe 9686 9695lowast 9689lowast 9695lowast 9695lowastPr 9269 9203 9444lowast 9481lowast 9445

Cardiac Arrhythmia Database

Ac 9764 9808lowast 9691 9766lowast 9585Fm 9435 9597lowast 9322 9377 9067Re 9607 9660lowast 9469 9544 9328Pr 9269 9535lowast 9179 9215 8821

Column 3C Weka

Ac 8577 8473 8429 8681lowast 8381Fm 7979 8103 8046 8358lowast 7986Re 8024 8142 8093 8395lowast 8037Pr 7935 8064 8000 8322lowast 7935

Glass Identification Database

Ac 8105 7539 7762 8155lowast 7557Fm 5685 5155 5049 5848lowast 4810Re 7985 5654 6647 7999lowast 5527Pr 4413 4738lowast 4070 4609lowast 4257

Heart Statlog

Ac 6920 7361lowast 7361lowast 7361lowast 7361lowastFm 7416 7719lowast 7719lowast 7719lowast 7719lowastRe 7581 7811lowast 7811lowast 7811lowast 7811lowastPr 7259 7259 7222 8629lowast 8074lowast

Iris Plants Database

Ac 9197 9598lowast 9687lowast 9687lowast 9732lowastFm 8801 9400lowast 9533lowast 9533lowast 9732lowastRe 8803 9401lowast 9533lowast 9533lowast 9604lowastPr 8800 9400lowast 9533lowast 9533lowast 9600lowast

Johns Hopkins University Ionosphere Database

Ac 6722 6717 6756lowast 6821lowast 6717Fm 7360 7355 7385lowast 7430lowast 7355Re 7858 7846 7877lowast 7905lowast 7846Pr 6923 6923 6951lowast 7008lowast 6923

Optical Recognition of Handwritten Digits

Ac 8451 8452lowast 8617lowast 8547lowast 8450Fm 1968 1954 3197lowast 2603lowast 1949Re 2515 2456 4163lowast 3302lowast 2459Pr 1617 1623lowast 2595lowast 2148lowast 1614

Pima Indians Diabetes Database

Ac 7777 7777 7751 7801lowast 7710Fm 8098 8098 8081 8110lowast 8053Re 8331 8331 8326 8327 8310Pr 7877 7877 7851 7903lowast 7812

Protein Localization Sites

Ac 9238 9205 9530lowast 9175 9185Fm 7944 7844 8861lowast 7801 7737Re 8163 8066 9006lowast 7928 7986Pr 7738 7633 8720lowast 7678 7503


Table 2 Continued


Vowel Context Data

Ac 8848 8881lowast 8868lowast 8877lowast 8775Fm 4226 4349lowast 4130 4196 3587Re 5306 5495lowast 4954 5397lowast 4359Pr 3511 3599lowast 3541lowast 3433 3048

Waveform Database Generator


Wine Recognition Data

Ac 9639 9477 9594 9748lowast 9751lowastFm 9514 9285 9445 9636lowast 9640Re 9535 9301 9453 9665lowast 9675lowastPr 9494 9269 9438 9606 9606

Wisconsin Breast Cancer


lowast)e Ac Fm Re and Pr values () higher than the standard approach

05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued


means of theAc Fm Re and Pr values Across 14 datasets theaverage of the Ac values of information gain (Ac) is 8550whereas the standard value is 8443 the average of Fmvalues of information gain (Fm) is 7605 whereas thestandard value is 7401 the average of Re values for in-formation gain (Re) is 8003 whereas the standard value is7850 and the average of Pr values for information gain(Pr) is 7323 whereas the standard value is 7071 Gainratio and correlation techniques also have higher averagevalues than standard however it is observed that reliefcannot obtain higher average values than standard for allmeasurements (specifically Fm and Re values are lower)

As a result the values in Tables 2 and 3 support the claimthat the most successful approach for determination of theweights of the features is weighted K-means++ clusteringwith information gain Furthermore in comparison with theother feature selection methods relief is not efficient fordetermining the weights

Traditional K-means++ clustering and four weightedK-means++ clustering approaches incorporating featureselection methods were compared in this analysis )enumber of clusters for each dataset was assumed to be theoriginal class number )e SSE values in Figure 3 show thatdistance metrics do not affect them significantly whereas itcan be seen clearly that all weighted clustering approachesincorporating feature selection methods produce fewer er-rors than traditional clustering for all datasets Moreover theweighted clustering decreased the SSE values by a largeproportion with information gain reducing SSE by 33 forthe ldquoBlocks Classificationrdquo dataset relief reducing SSE by75 for the ldquoCardiac Arrhythmia Databaserdquo dataset gainratio reducing SSE by 45 for the ldquoColumn 3C Wekardquodataset relief reducing SSE by 67 for the ldquoGlass Identi-fication Databaserdquo dataset gain ratio information gain andrelief reducing SSE by 73 for the ldquoHeart Statlogrdquo datasetgain ratio and relief reducing SSE by 35 for the ldquoIris PlantsDatabaserdquo dataset gain ratio reducing SSE by 54 for theldquoJohns Hopkins University Ionosphere Databaserdquo datasetgain ratio reducing SSE by 34 for the ldquoOptical Recognitionof Handwritten Digitsrdquo dataset relief reducing SSE by 56for the ldquoPima Indians Diabetes Databaserdquo dataset gain ratioreducing SSE by 42 for the ldquoProtein localization sitesrdquodataset relief reducing SSE by 77 for the ldquoVowel ContextDatardquo dataset information gain reducing SSE by 51 for theldquoWaveform Database Generatorrdquo dataset information gainreducing SSE by 47 for the ldquoWine Recognition Datardquodataset and relief reducing SSE by 76 for the ldquoWisconsinBreast Cancerrdquo dataset In particular relief gain ratio and

ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)

Figure 2 )e ℓ values for 8 algorithms and 10 datasets having more than 2 classes

Table 3 )e means of the Ac Fm Re and Pr values () for 14datasets and the standard and the weighted K-means++ clusteringapproaches

Means Standard Correlation Gain ratio Info gain ReliefAc 8443 8454lowast 8509lowast 8550lowastlowast 8448lowastFm 7401 7437lowast 7532lowast 7605lowastlowast 7349Re 7850 7742lowast 7969lowast 8003lowastlowast 7654Pr 7071 7148lowast 7252lowast 7323lowastlowast 7113lowastlowast)e Ac Fm Re and Pr values () higher than the standard approach lowastlowast)ehighest means of Ac Fm Re and Pr values ()


590

690

790

890

SSE

Standard Correlation Gain ratio Info gain Relief

EuclideanManhattanChebyshev

(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)


Standard Correlation Gain ratio Info gain Relief19

39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued


information gain are better than the other approaches indecreasing errors Moreover relief is the most successfulapproach to reducing SSE in proportion to its accuracyresults

In Figure 2 the approaches are compared with eachother separately for each dataset revealing that KS KP andB2FM have generally higher differences than the otherproposed approaches However because the ℓ values wereobtained between different ranges for each dataset theℓ values were normalized between 0 and 100 for each datasetthe average of each was calculated and the results wereobtained as the percentage value to enable clear comparison(Table 4) Finally compatibility values were computed as40 for KS 50 for KP and 39 for B2FM while muchhigher values of 96 for KQ 65 for WMC 80 for WMG79 for WMI and 69 for WMR were computed

)e results support the claim that KQ is the mostcompatible approach achieving 96 for all datasets Ad-ditionally the weighted approaches for KS KP and B2FMare more compatible than the traditional approaches Fur-thermore WMG and WMI have higher compatible valuesthan the other weighted approaches WMC and WMR Asa result because B2FM has a 37 compatibility resultbesides two most valuable features it can be claimed that theother features have high mapping efficiency as well

7 Conclusions and Future Work

In this paper five mapping approaches containing hybridalgorithms were proposed and presented in detail Oneinvolves K-means++ and QGA used together and the others

are weighted approaches based on four different featureselection methods (Pearsonrsquos correlation the gain ratioinformation gain and relief methods) According to theexperimental results by evaluation of DRV-P-MC as anoptimization problem mapping K-means++ centroids usingQGA emerges as the most successful approach with thelowest differences between the multivariate and 2D struc-tures for all datasets and the highest compatibility value(96) Conversely despite its reputation as themost popularmapping algorithm SOM did not perform as expected incompatibility tests in comparison with the proposed opti-mization algorithm However the weighted approachesbased on information gain gain ratio and relief providedmore consistent clusters than traditional clustering withmeans indicating approximately 50 lower SSE valuesAdditionally the experiments demonstrate that weightedclustering by using information gain provides the highestaverage accuracy (855) for all 14 datasets and achievesmore accurate placements on the map than PCA whilepreserving the most valuable features unlike PCA Asa result this study claims that the most successful hybridapproach for mapping the centroids is the integration ofweighted clustering with the information gain feature se-lection method and QGA



86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)

Figure 3 )e SSE values that were obtained by using Euclidean Manhattan and Chebyshev distance measurements for 14 datasets (a)Blocks Classification (b) Cardiac Arrhythmia Database (c) Column 3C Weka (d) Glass Identification Database (e) Heart Statlog (f ) IrisPlants Database (g) Johns Hopkins University Ionosphere Database (h) Optical Recognition of Handwritten Digits (i) Pima IndiansDiabetes Database (j) Protein Localization Sites (k) Vowel Context Data (l) Waveform Database Generator (m) Wine Recognition Data(n) Wisconsin Breast Cancer

Table 4 )e compatibility results () on 10 datasets for thetraditional and the proposed approaches

KS KP B2FM KQ WMC WMG WMI WMR40 50 37 96lowast 65 80 79 69lowast)e highest compatibility result ()


)is paper proposes novel approaches that are open toimprovement by the use of different hybrid structured al-gorithms For example all approaches in the paper havefocused on K-means++ clustering however the proposedapproaches can be adapted for other partitional clusteringalgorithms In addition it can be foreseen that for 3-di-mensional mapping the weighted approach based on in-formation gain will show higher performance than the otherapproaches because twomost valuable features are preservedfor two dimensions and the third one can be computed bythe weighted average of other features Moreover thisweighted approach offers utility by means of its consistencyand plain algorithmic structure for the visualization ofstream data because the updatable entries of clusteringfeatures can determine a summary of clustersrsquo statisticssimply Moreover the visualization of a large-scale datasetcan be achieved by using the proposed approaches after datasummarization by clustering with many clusters

)e proposed algorithms can be used extensively acrossa wide range of fields including medicine agricultural biologyeconomics and engineering sciences)ese fields have datasetspotentially containingmultidimensional structures If there arelots of instances in this multidimensional structure the size ofthe dataset mitigates against understanding the dataset andmakes decisions difficult Our algorithms presented in thispaper shrink the dataset size vertically by means of clusteringand horizontally by means of dimensional reduction

Notation

D DatasetF Set of the features in Df Length of FI Set of the instances in Dt Target attributen Length of IC Set of multivariate values of the centroidsE Set of 2D values of the centroidsk Number of centroidsM Matrix where the centroids are mappedr Row length of Mc Column length of Mℓ Difference among the multivariate centroids and their

obtained 2D projectionsS Matrix where the distances among the multivariate

centroids in C located in M are putT Matrix where the distances among 2D positions of the

centroids in C located in M are putZ Matrix where the values obtained after the subtraction

of S and T are putG Set of subcategories in a featureg Length of GB Set of subcategories in tb Length of B

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J P Cunningham and Z Ghahramani ldquoLinear dimensionalityreduction survey insights and generalizationsrdquo Journal ofMachine Learning Research vol 16 no 1 pp 2859ndash29002015

[2] H Abdi and L J Williams ldquoPrincipal component analysisrdquoWiley Interdisciplinary Reviews Computational Statisticsvol 2 no 4 pp 433ndash459 2010

[3] J Yang D Zhang A F Frangi and J Y Yang ldquoTwo-dimensional PCA a new approach to appearance-basedface representation and recognitionrdquo IEEE Transactions onPattern Analysis and Machine Intelligence vol 26 no 1pp 131ndash137 2004

[4] D Zhang and Z H Zhou ldquo(2D) 2PCA two-directionaltwo-dimensional PCA for efficient face representation andrecognitionrdquo Neurocomputing vol 69 no 1 pp 224ndash2312005

[5] A S N Curman M A Andresen and P J BrantinghamldquoCrime and place a longitudinal examination of street seg-ment patterns in Vancouver BCrdquo Journal of QuantitativeCriminology vol 31 no 1 pp 127ndash147 2015

[6] S Cicoria J Sherlock M Muniswamaiah and L ClarkeldquoClassification of titanic passenger data and chances ofsurviving the disasterrdquo in Proceedings of Student-FacultyResearch Day CSIS Pace University White Plains NYUSA May 2014

[7] S Hadi and K U J Sumedang ldquoClustering techniquesimplementation for detecting faces in digital imagerdquo Pen-gembangan Sistem Penciuman Elektronik Dalam MengenalAroma Campuran vol 1 pp 1ndash9 2006

[8] W Fang K K Lau M Lu et al ldquoParallel data mining ongraphics processorsrdquo Technical Report HKUST-CS08-07Hong Kong University of Science and Technology HongKong China 2008

[9] M Nitsche Continuous Clustering for a Daily News Sum-marization System Hamburg University of Applied SciencesHamburg Germany 2016

[10] V Wagh M X Doss D Sabour et al ldquoFam40b is requiredfor lineage commitment of murine embryonic stem cellsrdquo CellDeath and Disease vol 5 no 7 p e1320 2014

[11] N Raabe K Luebke and C Weihs ldquoKMCEDAM a newapproach for the visualization of K-means clustering resultsrdquoin Classification the Ubiquitous Challenge pp 200ndash207Springer Berlin Germany 2005

[12] A M De Silva and P H W Leong Grammar-Based FeatureGeneration for Time-Series Prediction Springer BerlinGermany 2015

[13] Y Zhang and G F Hepner ldquo)e dynamic-time-warping-based K-means++ clustering and its application in phenor-egion delineationrdquo International Journal of Remote Sensingvol 38 no 6 pp 1720ndash1736 2017

[14] T Sharma D Shokeen and D Mathur ldquoMultiple k-means++clustering of satellite image using Hadoop MapReduce andSparkrdquo 2016 httpsarxivorgabs160501802

[15] F Dzikrullah N A Setiawan and S Sulistyo ldquoImple-mentation of scalable K-means++ clustering for passengersrsquotemporal pattern analysis in public transportation system(BRT Trans Jogja case study)rdquo in Proceedings of IEEE


International Annual Engineering Seminar (InAES) pp 78ndash83 Daejeon Korea October 2016

[16] A P Nirmala and S Veni ldquoAn enhanced approach forperformance improvement using hybrid optimization algo-rithm with K-Means++ in a virtualized environmentrdquo IndianJournal of Science and Technology vol 9 no 48 2017

[17] L Wang S Li R Chen S Y Liu and J C Chen ldquoA seg-mentation and classification scheme for single tooth inmicroCT images based on 3D level set and K-means++rdquoComputerized Medical Imaging and Graphics vol 57pp 19ndash28 2017

[18] D Arthur and S Vassilvitskii ldquoK-means++ the advantages ofcareful seedingrdquo in Proceedings of ACM-SIAM Symposium onDiscrete Algorithms New Orleans LA USA pp 1027ndash1035July 2007

[19] B Bahmani B Moseley A Vattani R Kumar andS Vassilvitskii ldquoScalable K-means++rdquo Proceedings of theVLDB Endowment vol 5 no 7 pp 622ndash633 2012

[20] A Kapoor and A Singhal ldquoA comparative study of k-meansk-means++ and fuzzy c-means clustering algorithmsrdquo inProceedings of IEEE 3rd International Conference on Com-putational Intelligence amp Communication Technology (CICT)pp 1ndash6 Gwalior India February 2017

[21] T Kohonen ldquoEssentials of the self-organizing maprdquo NeuralNetworks vol 37 pp 52ndash65 2013

[22] T Kohonen ldquo)e self-organizing maprdquo Proceedings of theIEEE vol 78 no 9 pp 1464ndash1480 1990

[23] T Nanda B Sahoo and C Chatterjee ldquoEnhancing the ap-plicability of Kohonen self-organizing map (KSOM) esti-mator for gap-filling in hydrometeorological timeseries datardquoJournal of Hydrology vol 549 pp 133ndash147 2017

[24] J E Chu B Wang J Y Lee and K J Ha ldquoBoreal summerintraseasonal phases identified by nonlinear multivariateempirical orthogonal functionndashbased self-organizing map(ESOM) analysisrdquo Journal of Climate vol 30 no 10pp 3513ndash3528 2017

[25] A Voutilainen N Ruokostenpohja and T Valimaki ldquoAs-sociations across caregiver and care recipient symptoms self-organizing map and meta-analysisrdquo Gerontologist vol 58no 2 pp e138ndashe149 2017

[26] N Kanzaki T Kataoka R Etani K Sasaoka A Kanagawaand K Yamaoka ldquoAnalysis of liver damage from radon x-rayor alcohol treatments in mice using a self-organizing maprdquoJournal of Radiation Research vol 58 no 1 pp 33ndash40 2017

[27] W P Tsai S P Huang S T Cheng K T Shao andF J Chang ldquoA data-mining framework for exploring themulti-relation between fish species and water quality throughself-organizing maprdquo Science of the Total Environmentvol 579 pp 474ndash483 2017

[28] Y Dogan D Birant and A Kut ldquoSOM++ integration of self-organizingmap and k-means++ algorithmsrdquo in Proceedings ofInternational Workshop on Machine Learning and DataMining in Pattern Recognition pp 246ndash259 New York NYUSA July 2013

[29] F Viani A Polo E Giarola M Salucci and A MassaldquoPrincipal component analysis of CSI for the robust wirelessdetection of passive targetrdquo in Proceedings of IEEE In-ternational Applied Computational Electromagnetics SocietySymposium-Italy (ACES) pp 1-2 Firenze Italy March 2017

[30] D Tiwari A F Sherwani M Asjad and A Arora ldquoGreyrelational analysis coupled with principal component analysisfor optimization of the cyclic parameters of a solar-drivenorganic Rankine cyclerdquoGrey Systemseory and Applicationvol 7 no 2 pp 218ndash235 2017

[31] J M Hamill X T Zhao G Meszaros M R Bryce andM Arenz ldquoFast data sorting with modified principal componentanalysis to distinguish unique single molecular break junctiontrajectoriesrdquo 2017 httpsarxivorgabs170506161

[32] K Ishiyama T Kitawaki N Sugimoto et al ldquoPrincipalcomponent analysis uncovers cytomegalovirus-associated NKcell activation in Ph+ leukemia patients treated with dasati-nibrdquo Leukemia vol 31 no 1 pp 203ndash212 2017

[33] A D Halai A M Woollams and M A L Ralph ldquoUsingprincipal component analysis to capture individual differ-ences within a unified neuropsychological model of chronicpost-stroke aphasia revealing the unique neural correlates ofspeech fluency phonology and semanticsrdquo Cortex vol 86pp 275ndash289 2017

[34] S Wold K Esbensen and P Geladi ldquoPrincipal componentanalysisrdquo Chemometrics and Intelligent Laboratory Systemsvol 2 no 1ndash3 pp 37ndash52 1987

[35] T Hey ldquoQuantum computing an introductionrdquo IEEEComputing and Control Engineering Journal vol 10 no 3pp 105ndash112 1999

[36] A Malossini E Blanzieri and T Calarco ldquoQuantum geneticoptimizationrdquo IEEE Transactions on Evolutionary Compu-tation vol 12 no 2 pp 231ndash241 2008

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[38] Z Chen and W Zhou ldquoPath planning for a space-basedmanipulator system based on quantum genetic algorithmrdquoJournal of Robotics vol 2017 Article ID 3207950 10 pages2017

[39] G Guan and Y Lin ldquoImplementation of quantum-behavedgenetic algorithm in ship local structural optimal designrdquoAdvances in Mechanical Engineering vol 9 no 6 article1687814017707131 2017

[40] T Ning H Jin X Song and B Li ldquoAn improved quantumgenetic algorithm based on MAGTD for dynamic FJSPrdquoJournal of Ambient Intelligence and Humanized Computingvol 9 pp 1ndash10 2017

[41] D Konar S Bhattacharyya K Sharma S Sharma andS R Pradhan ldquoAn improved hybrid quantum-inspiredgenetic algorithm (HQIGA) for scheduling of real-timetask in multiprocessor systemrdquo Applied Soft Computingvol 53 pp 296ndash307 2017

[42] P Kumar D S Chauhan and V K Sehgal ldquoSelection ofevolutionary approach based hybrid data mining algorithmsfor decision support systems and business intelligencerdquo inProceedings of International Conference on Advances inComputing Communications and Informatics pp 1041ndash1046Chennai India August 2012

[43] N Singhal and M Ashraf ldquoPerformance enhancement ofclassification scheme in data mining using hybrid algorithmrdquoin Proceedings of IEEE International Conference ComputingCommunication amp Automation (ICCCA) pp 138ndash141Greater Noida India May 2015

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[46] B Athiyaman R Sahu and A Srivastava ldquoHybrid datamining algorithm an application to weather datardquo Journal ofIndian Research vol 1 no 4 pp 71ndash83 2013

[47] S Sahay S Khetarpal and T Pradhan ldquoHybrid data miningalgorithm in cloud computing using MapReduce frameworkrdquoin Proceedings of IEEE International Conference AdvancedCommunication Control and Computing Technologies(ICACCCT) pp 507ndash511 Ramanathapuram India 2016

[48] Z Yu L Li Y Gao et al ldquoHybrid clustering solution selectionstrategyrdquo Pattern Recognition vol 47 no 10 pp 3362ndash33752014

[49] P Sitek and J Wikarek ldquoA hybrid programming frameworkfor modelling and solving constraint satisfaction and opti-mization problemsrdquo Scientific Programming vol 2016pp 1ndash13 2016

[50] E Abdel-Maksoud M Elmogy and R Al-Awadi ldquoBraintumor segmentation based on a hybrid clustering techniquerdquoEgyptian Informatics Journal vol 16 no 1 pp 71ndash81 2015

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[53] R Jagtap ldquoA comparative study of utilization of single andhybrid data mining techniques in heart disease diagnosis andtreatment planrdquo International Journal of Computer Applica-tions vol 123 no 8 pp 1ndash6 2015

[54] M Dash H Liu and J Yao ldquoDimensionality reductionof unsupervised datardquo in Proceedings of IEEE NinthInternational Conference Tools with Artificial Intelligencepp 532ndash539 Newport Beach CA USA November 1997

[55] E Bingham and H Mannila ldquoRandom projection in di-mensionality reduction applications to image and text datardquoin Proceedings of the Seventh ACM SIGKDD InternationalConference on Knowledge Discovery and Data Miningpp 245ndash250 San Francisco CA USA August 2001

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[57] D Napoleon and S Pavalakodi ldquoA new method fordimensionality reduction using K-means clustering algorithmfor high dimensional data setrdquo International Journal ofComputer Applications vol 13 no 7 pp 41ndash46 2011

[58] S K Samudrala J Zola S Aluru and B GanapathysubramanianldquoParallel framework for dimensionality reduction of large-scaledatasetsrdquo Scientific Programming vol 2015 Article ID 18021412 pages 2015

[59] J P Cunningham and M Y Byron ldquoDimensionalityreduction for large-scale neural recordingsrdquo Nature Neuro-science vol 17 no 11 pp 1500ndash1509 2014

[60] L Demarchi F Canters C Cariou G Licciardi andJ C W Chan ldquoAssessing the performance of two unsu-pervised dimensionality reduction techniques on hyper-spectral APEX data for high resolution urban land-covermappingrdquo ISPRS Journal of Photogrammetry and RemoteSensing vol 87 pp 166ndash179 2014

[61] C Boutsidis A Zouzias M W Mahoney and P DrineasldquoRandomized dimensionality reduction for K-means clus-teringrdquo IEEE Transactions on Information eory vol 61no 2 pp 1045ndash1062 2015

[62] A T Azar and A E Hassanien ldquoDimensionality reduction ofmedical big data using neural-fuzzy classifierrdquo Soft Com-puting vol 19 no 4 pp 1115ndash1127 2015

[63] M B Cohen S Elder C Musco C Musco and M PersuldquoDimensionality reduction for K-means clustering and lowrank approximationrdquo in Proceedings of the Forty-SeventhAnnual ACM Symposium on eory of Computing pp 163ndash172 Portland OR USA June 2015

[64] X Zhao F Nie S Wang J Guo P Xu and X ChenldquoUnsupervised 2D dimensionality reduction with adaptivestructure learningrdquo Neural Computation vol 29 no 5pp 1352ndash1374 2017

[65] S Sharifzadeh A Ghodsi L H Clemmensen andB K Ersboslashll ldquoSparse supervised principal componentanalysis (SSPCA) for dimension reduction and variableselectionrdquo Engineering Applications of Artificial Intelligencevol 65 pp 168ndash77 2017

[66] Z Yu X Zhu H S Wong J You J Zhang and G HanldquoDistribution-based cluster structure selectionrdquo IEEETransactions on Cybernetics vol 47 no 11 pp 3554ndash35672017

[67] Z Yu and H S Wong ldquoQuantization-based clusteringalgorithmrdquo Pattern Recognition vol 43 no 8 pp 2698ndash27112010

[68] Z Wang Z Yu C P Chen et al ldquoClustering by localgravitationrdquo IEEE Transactions on Cybernetics vol 48 no 5pp 1383ndash1396 2018

[69] Z Yu L Li J Liu J Zhang and G Han ldquoAdaptive noiseimmune cluster ensemble using affinity propagationrdquo IEEETransactions on Knowledge and Data Engineering vol 27no 12 pp 3176ndash3189 2015

[70] Y Dogan D Birant and A Kut ldquoA new approach forweighted clustering using decision treerdquo in Proceedings ofInternational Conference on INISTA pp 54ndash58 IstanbulTurkey September 2011

[71] M Hall E Frank G Holmes B Pfahringer P Reutemannand I H Witten ldquo)e WEKA data mining software anupdaterdquo ACM SIGKDD Explorations Newsletter vol 11 no 1pp 10ndash18 2009


Computer Games Technology

International Journal of

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Journal ofEngineeringVolume 2018

Advances in

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ReconfigurableComputing



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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Human-ComputerInteraction

Advances in


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Submit your manuscripts atwwwhindawicom

As a result PCA is not typically successful at mappingthe dataset in 2D except in image representation andfacial recognition studies [3 4] Another difficulty isthat even if a dimensional reduction is applied the vi-sualization of all instances in a large dataset causes storageand performance problems To address this problemclustering algorithms can be used for data summarizationand the visualization methods applied afterwards K-Meansis the most-used clustering algorithm however K-meanssubmits only the high-dimensional centroids without anyvisualization and without any dimensional reductionoperation

Literature records three approaches to the visualizationof K-means )e first approach involves mapping for twoselected features from a multivariate dataset (MS2Fs)Curman et al have clustered the coordinate data inVancouver using K-means and presented them on a map[5] Cicoria et al have clustered ldquoTitanic Passenger Datardquousing K-means and printed the survival information thatwas the goal of the study to different 2D maps according tofeatures [6] an obtained binary image was clustered ontoa 2D interface using K-means for face detection in a studyby Hadi and Sumedang [7] and lastly Fang et al imple-mented software to visualize a dataset according to twoselected features [8] )e second approach is visualizationthat takes place after K-means clustering of the dataset isdimensionally reduced by PCA Nitsche has used this typeof visualization for document clustering [9] and Waghet al have used it for a molecular biological dataset [10])e third approach is the visualization of the datasetclustered by K-means in conjunction with an approach likeneighborhood method location in the self-organizing map(SOM) technique used by Raabe et al to implement a newapproach to show K-means clustering with a novel tech-nique locating the centroids on a 2D interface [11] SOM isa successful mapping and clustering algorithm neverthe-less it relies on the map-size parameter as the clusternumber and if it runs with a small map-size value allcentroids of the clusters are located contiguously not attheir original distances according to the high-dimensionalstructure

In our study new approaches are proposed to visu-alize the centroids of the clusters on a 2D map preservingthe original distances in the high-dimensional structure)ese different approaches are implemented as hybridalgorithms using K-means++ (an improved version ofK-means) SOM++ (an improved version of SOM)and the quantum genetic algorithm (QGA) QGA wasselected for use in our hybrid solutions because di-mensionality reduction and visualization problems formultivariate centroids (DRV-P-MC) are also an optimi-zation problem

)e contributions of this study are threefold firstclustering by K-means++ then mapping the centroids ontoa 2D interface using QGA and evaluating the success of thismethod A heuristic approach is proposed for DRV-P-MCand the aim is to avoid the drawback of locating the clusterscontiguously as in the traditional SOM++ Mapping thecentroids onto a 2D interface using SOM++ and evaluating

the success of this approach are performed to enablecomparison Second the usage of four major feature se-lection methods was mentioned in the paper by De Silva andLeong [12] specifically relief information gain gain ratioand correlation )e aim is to preserve the most valuablefeature and to evaluate it as the X axis Additionally the Yaxis is obtained by a weighted calculation using the co-efficient values returned from these feature selectionmethods )is provides an alternative to PCA for generatinga 2D dataset that avoids the PCA drawback of losing themost valuable feature )en clustering the datasets byK-means++ and mapping the centroids by these novelapproaches separately onto a 2D interface are performedMoreover mapping the centroids onto a 2D interface usingPCA and evaluating the success of this approach for com-parative purposes are performed )ird a versatile tool isimplemented with the capability to select the desired filealgorithm normalization type distance metric and sizeof the 2D map to calculate successes across six differ-ent metrics ldquosum of the square errorrdquo ldquoprecisionrdquo ldquorecallrdquoldquof-measurerdquo ldquoaccuracyrdquo and ldquodifference between multi-variate and 2D structuresrdquo )ese metrics are formulated indetail in Problem Definition

In literature generally MS2Fs has been the preferredmethod to manually determine the relations betweenfeatures Our tool is not only built on MS2Fs but alsocontains another algorithm that maps the centroids byusing information gain for comparison with our novelapproaches It assumes that the X axis is the most valuablefeature followed by the Y axis according to informationgain scores

)is paper details our study in seven sections DRV-P-MC is defined in detail in Section 2 related works providingguidance on DRV-P-MC including traditional algorithmshybrid approaches and the notion of dimensionality re-duction are submitted in Section 3 in Sections 4 and 5 thereorganized traditional approaches and the proposed al-gorithms are formulated and presented in detail the ex-perimental studies performed by using 14 datasets withdifferent characteristics and their accuracy results are givenin Section 6 and finally Section 7 presents conclusionsabout the proposed methods

2 Problem Definition

Considering that K-means++ is the major clustering algo-rithm used in this sort of problem the significance of patternvisualization in decision-making is clear However the vi-sualization requirement for the centroids of K-means++uncovers the need for DRV-P-MC since the centroidsreturned and the elements in them are presented by thetraditional K-means++ irrespective of any relation amongthe centroids )e aim in solving this problem is to map thecentroids onto a 2D interface by attaining the minimumdifference among the multivariate centroids and their ob-tained 2D projections

DRV-P-MC can be summarized as obtaining E from CTo detect whether the solution E of this problem is optimalor not ℓ must be measured as zero or a value close to zero




S⟵ 1113944k

i11113944

k

ji+1Sij

1113944

f


2

11139741113972

S⟵ 1113944k

i01113944

k

ji+1S[i j]

S[i j]

S[mi mj]

(1)


T⟵ 1113944k

i11113944

k

ji+1Tij

1113944

2


2

11139741113972

T⟵ 1113944k

i01113944

k

ji+1T[i j]

T[i j]

T[mi mj]

(2)





M







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


S

0 01 09 11

0 0 03 09

0 0 0 05

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 17 28 112

0 0 22 53

0 0 0 42

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S

0 x 9x 11x

0 0 3x 9x

0 0 0 5x

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 y 16y 65y

0 0 12y 31y

0 0 0 24y

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArr

S

0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)



Z⟵ 1113944k

i11113944

k


11138681113868111386811138681113868

11138681113868111386811138681113868

ℓ 1113944k

i11113944

k

ji+1Zij

(5)




0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArrZ

0 0 74 55

0 0 18 59

0 0 0 26

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

3 Related Works






















h expminusd2

2σ21113888 1113889

ω⟵ 1113944

f


(7)


1113944

k

i11113944

f






Lx LlowastLT

Ly LT lowastL(9)




L⟵1113944n


Ly⟵1113944

n

i1L

T lowastL

(10)







CPCA⟵ 1113944n

i11113944

2


CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (13)








i


i






FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






S⟵ 1113944k

i11113944

k

ji+1Sij

1113944

f


2

11139741113972

S⟵ 1113944k

i01113944

k

ji+1S[i j]

S[i j]

S[mi mj]

(1)


T⟵ 1113944k

i11113944

k

ji+1Tij

1113944

2


2

11139741113972

T⟵ 1113944k

i01113944

k

ji+1T[i j]

T[i j]

T[mi mj]

(2)





M







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


S

0 01 09 11

0 0 03 09

0 0 0 05

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 17 28 112

0 0 22 53

0 0 0 42

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S

0 x 9x 11x

0 0 3x 9x

0 0 0 5x

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 y 16y 65y

0 0 12y 31y

0 0 0 24y

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArr

S

0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)



Z⟵ 1113944k

i11113944

k


11138681113868111386811138681113868

11138681113868111386811138681113868

ℓ 1113944k

i11113944

k

ji+1Zij

(5)




0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArrZ

0 0 74 55

0 0 18 59

0 0 0 26

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

3 Related Works






















h expminusd2

2σ21113888 1113889

ω⟵ 1113944

f


(7)


1113944

k

i11113944

f






Lx LlowastLT

Ly LT lowastL(9)




L⟵1113944n


Ly⟵1113944

n

i1L

T lowastL

(10)







CPCA⟵ 1113944n

i11113944

2


CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (13)








i


i






FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






0 1 9 11

0 0 3 9

0 0 0 5

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus

0 1 16 65

0 0 12 31

0 0 0 24

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

rArrZ

0 0 74 55

0 0 18 59

0 0 0 26

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

3 Related Works






















h expminusd2

2σ21113888 1113889

ω⟵ 1113944

f


(7)


1113944

k

i11113944

f






Lx LlowastLT

Ly LT lowastL(9)




L⟵1113944n


Ly⟵1113944

n

i1L

T lowastL

(10)







CPCA⟵ 1113944n

i11113944

2


CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (13)








i


i






FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in

















h expminusd2

2σ21113888 1113889

ω⟵ 1113944

f


(7)


1113944

k

i11113944

f






Lx LlowastLT

Ly LT lowastL(9)




L⟵1113944n


Ly⟵1113944

n

i1L

T lowastL

(10)







CPCA⟵ 1113944n

i11113944

2


CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (13)








i


i






FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




CPCA⟵ 1113944n

i11113944

2


CPCA⟵ 1113944k

i11113944

2

j1CPCA(i j)

CPCA(i j)

n

(12)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (13)








i


i






FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




FC⟵ 1113944k

i1C(i fc) (14)

SC⟵ 1113944k

i1C(i sc) (15)


E⟵ 1113944k

i11113944

2

j1E(i j)


maxj minusminj1113872 1113873 (16)







2




Mprime⟵ 1113944r

i11113944

c

j

Mijprime 0

Mprime⟵ 1113944k

i1Mrc rrprime

i rngt αrcrr1113868111386811138681113868

11138681113868111386811138682

i it 0



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)










0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(18)

Δz





0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(19)

αβ



αprime

βprime1113890 1113891 (20)







CC⟵ 1113944

f

i1



nj1tj

nlowast1113936nj1I

2ij minus 1113936

nj1Iij1113872 1113873

21113876 1113877lowast nlowast1113936

nj1t

2j minus 1113936

nj1tj1113872 1113873

21113876 1113877

1113970

CC⟵ 1113944

f

i1CCi

CCi


maxccCCi

maxcc lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)
















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in


















GR⟵minus1113936b

i1P Bi( 1113857 log P Bi( 1113857 + 1113936bi11113936

g

j1P Gi( 11138571113936bi1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875


GR⟵1113944

f

i1GRi

GRi

mingr mingr gt 0

maxgrGRi

maxgr lt 0



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)








For i 1 kWCi WCiwc








For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







For i 1 kWCi WCiwc







For i 1 kWCi WCiwc




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




IG⟵minus1113944b

i1P Bi( 1113857 log P Bi( 1113857

+ 1113944b

i11113944

g

j1P Gi( 1113857 1113944

b

i1P Bi Gj

111386811138681113868111386811138681113874 1113875 log P Bi Gj

111386811138681113868111386811138681113874 1113875

IG⟵1113944

f

i1IGi

IGi

minIG minIG gt 0

maxIGIGi

maxIG lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)



RF⟵1113944

f

i1RFi 0

RF⟵1113944n

i11113944

f


11138681113868111386811138681113868

11138681113868111386811138681113868 + Imj minusRi

11138681113868111386811138681113868

11138681113868111386811138681113868

RF⟵1113944

f

i1RFi

RFminrf

minrf gt 0

maxrfRFi

maxrf lt 0


lowast 09 + 01 else

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)





dij

1113944

f


2

11139741113972

(25)

dij 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868 (26)

dij max 1113944

f

a1Iia minus Ija

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠ (27)







FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





FCi CifcFor i 1 k



For i 1 kWCi WCiwc







NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




NV FVminus FVmin


NV FVminus micro

b (29)

NV FV10τ

(30)

NV FV

NVmax (31)





Ac Tp + Tn

Tp + Tn + Fp + Fn (32)


Pr + Re (33)

Re Tp

Tp + Tn (34)

Pr Tp

Tp + Fp (35)



SSE 1113944k

i11113944

f


2 (36)














Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in












Column 3C Weka




Heart Statlog













Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




Table 2 Continued


Vowel Context Data









05

1015202530


ℓ val

ues

KSKPB2FM

KQWMCWMG

WMIWMR

(a)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(b)

ℓ val

ues

0010203040506

Column 3C Weka

KSKPB2FM

KQWMCWMG

WMIWMR

(c)

ℓ val

ues

0

18

36

54

72

90


KSKPB2FM

KQWMCWMG

WMIWMR

(d)

ℓ val

ues

0

04

08

12

16

2


KSKPB2FM

KQWMCWMG

WMIWMR

(e)

ℓ val

ues

04590

135180225


KSKPB2FM

KQWMCWMG

WMIWMR

(f )

Figure 2 Continued





ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







ℓ val

ues

0

18

36

54

72


KSKPB2FM

KQWMCWMG

WMIWMR

(g)

ℓ val

ues

050

100150200250300

Vowel Context Data

KSKPB2FM

KQWMCWMG

WMIWMR

(h)

ℓ val

ues

0

5

10

15

20


KSKPB2FM

KQWMCWMG

WMIWMR

(i)

ℓ val

ues

0

01

02

03

04


KSKPB2FM

KQWMCWMG

WMIWMR

(j)





590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




590

690

790

890

SSE



(a)

200

400

600

800



SSE

(b)



36

46

56

66

SSE

(c)



39

59

SSE

(d)



71

171

271

SSE

(e)



23

28SS

E

(f )



170

270

370

SSE

(g)



6690

7690

8690

SSE

(h)



115

165

215

265

SSE

(i)



50

60

70

SSE

(j)

Figure 3 Continued










86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in












86

186

286

386SS

E

(k)



2866

3866

SSE

(l)



45

65

85

SSE

(m)



192

292

392

SSE

(n)







Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






Notation






Data Availability




References

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




NovelTwo-DimensionalVisualizationApproachesfor...

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