a modular takagi-sugeno-kang (tsk) system based on a modi

Scientia Iranica E (2021) 28(4), 2342{2360

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

A modular Takagi-Sugeno-Kang (TSK) system basedon a modi�ed hybrid soft clustering for stock selection

S. Mousavia, A. Esfahanipourb;�, and M.H. Fazel Zarandib

a. Department of Industrial Engineering, Meybod University, Meybod, Iran.b. Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, P.O. Box 15914,

Iran.

Received 23 November 2018; received in revised form 8 September 2019; accepted 20 October 2019

KEYWORDSIntelligent modularsystems;Ensemble learning;Hybrid rough-fuzzyclustering;TSK fuzzy rule-basedsystem;Stock selection;Tehran StockExchange (TSE).

Abstract. This study presents a new hybrid intelligent system with ensemble learningfor stock selection using the fundamental information of companies. The system uses theselected �nancial ratios of each company as input variables and ranks the candidate stocks.Due to the di�erent characteristics of the companies from di�erent activity sectors, modularsystem for stock selection may show a better performance than an individual system.Here, a hybrid soft clustering algorithm was proposed to eliminate the noise and partitionthe input dataset into more homogeneous overlapped subsets. The proposed clusteringalgorithm bene�ts from the strengths of the fuzzy, possibilistic and rough clustering todevelop a modular system. An individual Takagi-Sugeno-Kang (TSK) system was extractedfrom each subset using an arti�cial neural network and genetic algorithm. To integrate theoutputs of the individual TSK systems, a new weighted ensemble strategy was proposed.The performance of the proposed system was evaluated among 150 companies listed onTehran Stock Exchange (TSE) regarding information coe�cient, classi�cation accuracy,and appreciation in stock price. The experimental results show that the proposed modularTSK system signi�cantly outperforms the single TSK system as well as other ensemblemodels using di�erent decomposition and combination strategies.© 2021 Sharif University of Technology. All rights reserved.

1. Introduction

The problem of fund allocation involves two stages:asset selection and asset allocation. In the �rststage, the objective is to select some attractive andvaluable assets as the potential candidates for portfoliocomposition. In the second stage, the objective isto determine portfolio weights of the selected assetsto achieve a series of risk-return considerations [1].

*. Corresponding author. Tel.: +98 21 6454-5300;Fax: +98 21 6695-4569E-mail addresses: [email protected] (S. Mousavi);[email protected] (A. Esfahanipour); [email protected](M.H. Fazel Zarandi)

doi: 10.24200/sci.2019.52323.2661

Similarly, in stock portfolio management, one shouldselect a universe of stocks before running a stock port-folio optimization model to determine optimal portfolioweights [2]. Those selected stocks have the best chancesof capital appreciation in a long- or intermediate-termhorizon.

There are two approaches widely used by aca-demicians and market professionals for decision-makingin stock exchange: technical analysis and fundamen-tal analysis. The fundamental analysis involves adetailed study of a company's �nancial status usingthe �nancial ratios and other fundamentals of thecompany to predict the future stock price movements.In the fundamental analysis, the main concern isthe company's �nancial health. Traders often usethis approach to predict the stock price over a long-

S. Mousavi et al./Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 2342{2360 2343

term investment horizon. On the other hand, thetechnical analysis uses the data related to the pastbehavior of a stock price and volume data series toforecast the future. Traders use this approach in short-term investment horizons. They deal with the stocktiming rather than the company's �nancial health [3{7]. Since the fundamentals have a stronger relationshipwith the price movement in the longer horizons, thestock selection stage should be designed based on thefundamental analysis.

This study tries to develop a stock selectionsystem based on fundamental analysis. The systemuses the selected �nancial ratios and fundamental dataof each company as the input variables and ranksthe candidate stocks based on the fundamental data.However, the future performance of the companies mayfollow di�erent patterns due to di�erent fundamentalcharacteristics as well as activity sectors of the com-panies [8,9]. For example, the companies with highinventory turnover usually have a lower current ratiothan the companies in other activity sectors. A lowcurrent ratio is not always an indicator of poor liquidityperformance and should be compared with the currentratios of the other companies with similar inventoryturnover. Therefore, it seems that a modular systemfor stock selection may exhibit better performance thanan individual system [10]. Furthermore, based on theprinciple of divide and conquer, the complexity of thewhole data space is reduced by modularity, which leadsto more homogeneous data spaces [11].

In general, three main steps should be done todevelop a modular system with ensembles. In the�rst step, the training dataset is partitioned into somesmaller data regions. In the second step, an individuallearner is developed for each data region, separately.In the third step, the outputs of the individual learnersare combined to determine the �nal output of themodular system using an ensemble strategy. In thisstudy, a hybrid Rough-Fuzzy Noise Rejection Clus-tering (RFNRC) algorithm is proposed to determinethe overlapping data regions of the modular systemand remove the noise data, simultaneously. Then,an individual Takagi-Sugeno-Kang (TSK) fuzzy rule-based system is developed for stock selection in eachdata region separately. Finally, the outputs of theindividual systems are combined to derive the ultimateresult using our proposed ensemble strategy.

The main purpose of this paper is to construct anaccurate and interpretable stock selection system forportfolio managers. The system ranks the universe ofstocks and selects a set of stocks that are likely to havethe best chances of capital appreciation in the subse-quent period. For this purpose, we propose an ensemblelearning model to develop a modular stock selectionsystem based on our proposed clustering. Here, we de-scribe the novelties of this study from two perspectives.

The �rst one regards the applied method to modularizethe system and the ensemble strategy. This studyproposes a hybrid RFNRC algorithm to partition anoisy data set into some overlapping partitions withoutnoise and outliers. The proposed model also developsa new weighted ensemble strategy to aggregate theoutputs of the modules. The second one relates tothe development of a modular system for the stockselection problem through the ranking of the stocks.To the best of the authors' knowledge, this is the �rststudy that develops a TSK system for stock selectionproblem which applies an ensemble learning model.

The rest of the paper is organized as follows.Section 2 provides a brief review of the previous worksrelated to our study. Section 3 explains the proposedalgorithm for hybrid rough-fuzzy noise-rejection clus-tering. Section 4 presents a comprehensive descriptionof the proposed ensemble learning model. Section 5describes the implementation of the proposed model todevelop a modular system for stock selection on TehranStock Exchange (TSE) as well as the computationalresults. Finally, Section 6 reports some concludingremarks.

2. Related works

According to the principle of divide and conquer,complex computational learning can be simpli�ed bydividing the learning task among some experts andthen, combining the solutions of the experts, which issaid to constitute a committee machine [10]. Modularsystems are one of the architectural types of committeemachines from the local accuracy perspective [12]. Inthe modular systems, separate learners are developedand applied to di�erent regions of the problem domain.The regions of interest are determined �rst, and anindividual learner is then developed for each region.With this architecture, an important issue is theidenti�cation of the best regions to be considered. Theregions can be de�ned based on expert opinion [13] orusing purely mechanical means like clustering [14].

2.1. Related works on clustering algorithmsMany researchers have used hard clustering to de-termine the data regions of a modular system [15{17]. However, the boundaries between the adjacentdata regions may be unclear, and a data instance maynot completely belong to only one cluster. Therefore,hard clustering is too restrictive in partitioning. Thesoft clustering algorithms using fuzzy and rough settheories are less restrictive than the hard clustering bypermitting an instance to belong to multiple clusters.While the fuzzy clustering can e�ciently handle theoverlapping clusters [18], it can be too descriptivewith potentially a list of possible memberships for anindividual object [19]. In this way, the data set may not

2344 S. Mousavi et al./Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 2342{2360

be partitioned into some homogenous data spaces withless complexity. Therefore, most of the previous studiesrelated to the development of the modular systemwith fuzzy clustering have applied traditional fuzzyclustering in which each sample is assigned to only onecluster based on maximum membership [20]. Roughclustering, as another soft clustering approach, allowsan object to belong to multiple clusters. In roughclustering, representation of the clusters is based onthe lower and upper bounds using rough set theoreticproperties. Lingras and West [21] believed that thelower and upper bound representation of a clusterwas more concise than the detailed and descriptivelist of membership values. They suggested Rough C-Means (RCM) clustering as the �rst rough clusteringalgorithm. In this algorithm, the clusters are repre-sented by the crisp lower and upper approximations.The lower approximation contains objects that aremembers of the cluster with certainty (probability = 1),while the upper approximation contains objects thatare members of the cluster with non-zero probability(probability > 0). However, the rough clusters do notdetermine the similarity and closeness of the instancesto the cluster prototypes [22].

In the last decade, intensive works have beendone for hybridization of rough and fuzzy clusteringto integrate the advantages of both fuzzy sets andrough sets [18,22{25]. Hybrid Rough-Fuzzy C-Means(RFCM) clustering was proposed by Mitra et al. in2006 for the �rst time [22]. In their proposed clustering,each cluster consisted of a fuzzy lower approximationand a fuzzy boundary. A hybrid clustering with thecrisp lower approximation and the fuzzy boundarieswas proposed by Maji and Pal [24]. Furthermore,a rough-fuzzy Possibilistic C-Means (PCM) clusteringwas extended to make the previous hybrid cluster-ing [24] robust in the presence of noise and outliers [18].Many research �elds bene�t from the applications ofhybrid rough-fuzzy clustering such as bioinformaticsand medical imaging [26,27] and text-graphics seg-mentation [28]. Recently, ensemble-based rough-fuzzyclustering has been extended for categorical data withdi�erent dissimilarity measures [29,30]. However, thementioned hybrid clustering algorithms su�er fromsome weak points that are explained in Section 3. Thisstudy proposes a new hybrid clustering algorithm thattries to overcome the weakness of these algorithms fordeveloping a modular system. The proposed algorithmbene�ts the strengths of the fuzzy, possibilistic andrough clustering approaches while it is subject to theirweak points for developing a modular system.

2.2. Related works on ensemble strategiesThe literature on ensemble learning has shown thatthe ensemble can outperform single predictors inmany cases [31{33]. Additionally, ensembles of ex-

pert systems have already been successfully appliedin �nancial forecasting [34{43]. There are di�erentensemble strategies in the literature including simplemajority vote, simple averaging, weighted averaging,reliability-based strategies, Bayesian methods, andstacking [44,45]. Among the mentioned ensemblestrategies, the simple majority vote and the simpleaveraging of the baseline classi�ers have had rel-atively poor performance in di�erent �elds includ-ing �nancial time series prediction [12,40]. On theother hand, weighted averaging and stacking strategieshave been applied with great success over the lastfew years [13,41,46{49]. In stacking, a high-levelbase learner is developed to combine the lower levelbase learners, while the base learners are combinedwith di�erent weights in the weighted averaging. Inthese ensemble approaches, the integration of thebase learners is done using a weighted least squaresalgorithm [46,48,49], generalized regression neural net-works [47], particle swarm optimization [41], and ge-netic fuzzy systems [13]. Lv et al. [48] found thatusing the fuzzy memberships for di�erent data parti-tions improved the accuracy of the ensemble model.Considering the results of the mentioned studies andour proposed clustering method, a new weighted en-semble was introduced that would use the rough fuzzymemberships of the data partitions.

2.3. Stock selection based on the fundamentalanalysis

Fundamental analysis has been widely used for stockselection in stock portfolio management. The stock se-lection models based on Fundamental analysis includePROMETHEE decision-making model [50], general-ized data envelopment analysis model [1], Multiple At-tribute Decision Making (MADM) model [51], Probitand Tobit based models [52], and also a continuous timemodel for active portfolio management [53]. However,soft computing models seem to be more appropriate formodeling the noisy, nonlinear and complex behaviorof the stock markets [54{58]. In the literature, thereare some studies on soft computing methods such asArti�cial Neural Networks (ANN) [3,59], evolutionaryalgorithms [60{63], support vector machines [64], andfuzzy logic [65{68] for stock selection problem. Thisstudy proposes a hybrid genetic fuzzy system to selectthe best stocks in the portfolio composition. TheTSK fuzzy rule-based systems have exhibited a goodcapability for modeling the nonlinear dynamic systemsin many �elds including the short-term stock trendprediction [69{71]. This paper intends to evaluate theperformance of TSK systems in the stock ranking andselection problem over the longer investment horizons.The structure and parameter identi�cation phases ofthe TSK systems are done using ANN and GeneticAlgorithms (GA), respectively. Table 1 compares this


Table 1. Position of this study among the related studies in the literature.

Reference Learning toolNo. of

fundamentalvariables

Handlenonlinearity

Fitness measureTypeof thesystem

Consideringfundamental

characteristicsof sectors

Beckeret al. [60]

Genetic programming 65 X Information coe�cientand spread

Crisp �

Quah [3] Arti�cial neural networks 11 X Classi�cation accuracyCrispand

fuzzy�

Huanget al. [65]

Genetic algorithm-fuzzy 12 � Return of top rankedstocks

Fuzzy �

Huang [95] Support vector regression-genetic algorithms

14 X Cumulative return ofthe selected stocks

Crisp �

Vanstoneet al. [59]

Arti�cial neural networks 4 XMax. percentagechange in price overnext 200 days

Crisp �

Parqueet al. [61]

Genetic networkprogramming

10 X Risk-adjusted returnof selected stocks

Crisp �

Ince [62] Genetic algorithm-case-based reasoning

7 X Classi�cation accuracy Crisp �

Yu et al.[64]

Support vector machines 20 XClassi�cation accuracy& the portfolioaccumulated return

Crisp �

Silva et al.[63]

Genetic algorithm 10 � Return and risk ofproposed portfolio

Crisp �

Shen andTzeng [66]

VIKOR, DANP, DEMATEL(decision-making trial andevaluation laboratory)

17 X Classi�cation accuracy Fuzzy �

Thisstudy

Genetic algorithm-arti�cial neural networks

36 X Information coe�cient

TSK typefuzzy

rule-basedsystem

X

study with the previous researches for stock selectionbased on the fundamental analysis using soft comput-ing methods.

3. The developed hybrid rough-fuzzynoise-rejection clustering algorithm

The hybrid rough-fuzzy clustering incorporates fuzzymembership values in the rough clustering framework.RFCM algorithm was proposed by Mitra et al. [22]for the �rst time. This algorithm partitions a setof N objects X = fx1; � � � ; xj ; � � � ; xNg into c roughclusters (Ui) with a fuzzy lower approximation and afuzzy boundary by minimizing the objective functionJRFCM as Eq. (1) subject to

Pci=1 uij = 1 for all

j = 1; 2; � � � ; N .

JRFCM =cXi=1

JCi;

JCi=

8><>:wlow�Ai+wbound�Bi; if AUi 6= �; BUi 6= �Ai; if AUi 6= �; BUi = �Bi; if AUi = �; BUi 6= �

Ai =X

xj2AUiumijkxj � vik2;

Bi =X

xj2BUiumijkxj � vik2; (1)

where vi is the center of cluster Ui, k � k is the distancenorm, uij is the membership of xj to cluster Ui, and1 � m <1 is the fuzzi�er in the fuzzy set theory. AUiand AUi are the lower and the upper approximationsof Ui and BUi = AUi � AUi denotes the boundaryregion of the rough cluster Ui. The terms Ai and Birepresent the weighted within-groups sum of squarederrors for the lower approximation and boundary of


rough clusters, respectively. The parameters wlowand wbound are the relative importance of the lowerapproximation and the boundary regions.

According to the de�nitions of lower approxima-tion and boundary of rough sets by Pawlak, if an objectis the member of a cluster's lower approximation, itde�nitely belongs to that cluster and it cannot be amember of its boundary or the other clusters [72].

If xj 2 AUi Then xj =2 BUk;8 k and xj =2 AUk; 8 k 6= i:

Maji and Pal [24] claimed that based on these de�ni-tions, the objects in the lower approximation shouldhave a similar in uence on only their correspondingcluster regardless of their similarity with their cor-responding clusters and the other clusters. Theyproposed a hybrid rough-fuzzy clustering with the crisplower approximation and the fuzzy boundaries. In thiscase, they reduced Ai to Eq. (2):

Ai =X

xj2AUikxj � vik2: (2)

They incorporated PCM into their previous modelto develop a more robust clustering algorithm in thepresence of noise and outliers [18]. In their proposedrough-fuzzy PCM algorithm, they calculated Ai similarto their previous work and changed Bi as Eq. (3):

Bi =X

xj2BUifa(�ij)m1 + b(�ij)m2gkxj � vik2

+ �iX

xj2BUi(1� �ij)m2 ; (3)

where �ij is the probabilistic membership of xj toUi as that in Fuzzy C-Means (FCM) and �ij is thepossibilistic membership as in the PCM. The constantsa and b determine the relative importance of the prob-abilistic and possibilistic memberships, respectively.The objective function of their proposed clusteringalgorithm is minimized when:

�ij =

cX

k=1

� kxj � vik2kxj � vkk2

� 2m1�1

!�1

; (4)

�ij =1

1 +�bkxj�vik

�i

� 1m2�1

: (5)

Also, a Rough Possibilistic Type-2 Fuzzy C-Means(RPT2FCM) clustering algorithm was proposed bySarkar et al. [73]. The RPT2FCM algorithm is quitesimilar to the rough-fuzzy PCM algorithm proposedby Maji and Pal [18]. The only di�erence between

them lies in probabilistic memberships. In [73], theprobabilistic memberships of Eq. (3) were type-2 fuzzymembership values to handle some other various subtleuncertainties in the overlapping areas.

Although the lower approximation members ofrough clusters de�nitely belong to their correspondingclusters, it is unreasonable to impose the same weightfor all objects of a lower approximation [74]. We believethat di�erent objects of a lower approximation shouldhave di�erent weights based on the proximity to theircorresponding cluster prototypes regardless of otherprototypes. This is consistent with the above de�nitionof lower approximation by Pawlak [72]. In this paper,we propose a hybrid RFNRC algorithm to resolvesome of the drawbacks of the previous hybrid c-meansalgorithms. It incorporates the Fuzzy Noise RejectionClustering (FNRC) [75] into the RCM framework. TheFNRC utilizes FCM and PCM to introduce a morerobust clustering algorithm in the presence of noise andoutliers.

In our proposed algorithm, di�erent objects havedi�erent weights in determining their prototypes sim-ilar to [22]. However, unlike their work, the weightsonly depend on the distance of the objects from theircorresponding prototypes which is more consistent withthe rough set theory. The main steps of the proposedRFNRC algorithm are as follows. Steps 1{3 aredesigned to de�ne the suitable weighting exponent, thenumber of clusters, and the initial cluster centers asthe preprocessing steps of RFNRC. The fourth stepdetermines the fuzzy clusters. Steps 5{7 are related tonoise rejection from the dataset. In the 8th step, thePCM membership values are calculated and Steps 9and 10 determine the rough clusters.

Step 1. De�ne the suitable weighting exponent (m).The weighting exponent should be selected far fromits both extreme ends to guarantee that the clustervalidity index in the next step indicates the optimumnumber of clusters. According to [76], the suitableweight exponent is a value that makes the tracing ofthe fuzzy total scatter matrix (ST ) equal to z=2.

ST =NXj=1

cXi=1

(uij)m!

(xj � v) (xj � v)T ; (6)

z = trace

0@ NXj=1

240@xj � 1N

NXj=1

xj

1A0@xj � 1

N

NXj=1

xj

1AT3751CA ; (7)

where v is the fuzzy total mean vector of the datasetconsidering the FCM-based membership values.


Step 2. Determine the optimum number of clusters(C) through the original FCM so that the clustervalidity index Eq. (8) can be minimized. This indexdetermines the optimum number of clusters thatmaximizes within clusters compactness and betweenclusters separation.

Scs =NXj=1

cXi=1

(uij)m�kxj � vik2 � kvi � vk2� : (8)

Step 3. Assign the initial cluster centers by theagglomerative hierarchical clustering algorithm. Thisclustering algorithm puts each of the n data instancesin an individual cluster. Then, two or more clustersare merged using a matrix of dissimilarities until therequired number of clusters (C) is reached. This stepprevents our proposed algorithm from converging toa local extreme.Step 4. Identify the initial fuzzy cluster prototypesusing the original FCM.(a) Compute the matrix of membership degrees:

uij =

CXk=1

� kxj � vik2kxj � vkk2

� 2m�1

!�1

; (9)

(b) Update the cluster centers:

vi =PNj=1(uij)mxjPNj=1(uij)m

; (10)

(c) Repeat (a) and (b) until reaching convergence incluster centers, i.e., v(t)

i � v(t�1)i < ", where t is

the iteration number of the FCM algorithm.Step 5. Calculate the resolution parameter of PCM(�i). The value of �i determines the distance that themembership value of a point in the cluster i becomes0.5 and it is chosen based on the desired \bandwidth"of the possibilistic membership distribution for eachcluster [77]. It is assumed that the data in eachcluster follows a Gaussian distribution and kxj �vik=�i has a chi-square distribution, with degrees offreedom equivalent to the number of features in eachdata instance. Therefore, the resolution parametercan be calculated as follows [75]:

�i =

median (kxj � vik)xj 2 Ui�2

0:5; (11)

where �2 is the chi-square value.Step 6. Calculate the cuto� distance (uFC cut2) todetect the noise and outliers [75].

uFC cut2 = �i�2z; (12)

where z is the percentage of inliers in the data. Thenumber of outliers is estimated based on W index.

Wj =CXi=1

kxj � vik: (13)

This index sums the distance of the data instanceto all cluster centers. The data instances with largevalues of Wj are considered as outliers. The thresholdfor outliers depends on the upper and lower bounds ofthe data and is selected according to the trace of Windex. This step is designed to �nd the outliers, i.e.,the data instances which are too far from all clustercenters.Step 7. Remove the noise data and outliers. If kxj�vik > uFC cut2, then the data instance is recognizedas noise and it takes a zero membership to the cluster.Step 8. Compute the membership matrix of theremaining data using PCM membership value [77]:

uij =1

1 +�kxj�vik

�i

� 1m�1

: (14)

Step 9. Assign each data instance to the lowerapproximation of a cluster or the boundaries ofmultiple clusters through the following procedure:(a) Assign the instance (xj) to the upper bound

of the cluster k(AUk) based on the maximummembership:

xj 2 AUk; k = argi maxi=1;2;�� ;C(uij):(15)

The instance should be assigned to the upperbound and boundary of two or more clusters inthe case of ties in the maximum membership;

(b) For each cluster i, i = 1; 2; � � � ; C, i 6= k, if ukj �uij < �, then xj 2 AUi and xj 2 BUi, where� is a small threshold value that determines theoverlapping degree of the adjacent clusters;

(c) If xj =2 AUi, i = 1; 2; � � � ; C, i 6= k, then xj 2AUk, else xj 2 BUk.

Step 10. Compute the new cluster centers inEq. (16):

vi=

8><>:wlow�C1+wbound�D1; if AUi 6= ;; BUi 6= ;C1; if AUi 6= ;; BUi = ;D1; if AUi = ;; BUi 6= ;

C1 =

Pxj2(AUi) u

mijxjP

xj2(AUi) umij

;

D1 =

Pxj2(BUi) u

mijxjP

xj2(BUi) umij

: (16)

According to the rough set theory, the objects of


the lower approximation should have much morein uence on their cluster prototypes. Therefore, wlowshould be much more than wbound, i.e.:

0 < wbound < wlow < 1; wlow + wbound = 1:

Step 11. Repeat Steps 8{10 until convergence isreached, i.e., there are no more new assignments.

The strengths of our proposed RFNRC algorithmare given in the form of six aspects. First, basedon the compatibility with the centroid point of view,possibilistic memberships of PCM correspond moreclosely to the notion of typicality [77]. Unlike FCM,there is no constraint on the memberships of PCM (i.e.,PCi=1 �ij = 1, 8 j). Therefore, the prototypes of PCM

are attracted toward dense regions in the feature spaceregardless of the locations of the other prototypes.Second, using FCM and agglomerative hierarchicalclustering at the �rst steps avoids the problem ofthe coincident clusters of PCM. The FCM and PCMalgorithms have been previously integrated to avoidthe problems of noise sensitivity of the FCM and thecoincident clusters of PCM [18]. Third, Steps 5{7 of theproposed algorithm make the algorithm more robustin the presence of noise and outliers. The powerfulability of these steps in noise rejection was con�rmedby Melek et al. [75]. Our proposed algorithm removesthe noise and outliers from the cluster members and,therefore the outliers do not a�ect the learning processof the modules in the modular system. Fourth, the c-means algorithms with random initial centers alwaysconverge to a local extreme [78]. Our proposedRFNRC algorithm has overcome this problem usingagglomerative hierarchical clustering algorithm. Fifth,it uses two preliminary steps to identify the suitableweight exponent and the optimum number of clusterswhich make our algorithm more e�cient. Above all, theprevious hybrid clustering algorithms allow only twooverlapping clusters [18,22,24]. However, three or moreoverlapping clusters are possible in clustering of realdata sets. Step 9 (parts (a) and (b)) of our proposedRFNRC handles multiple overlapping clusters as wellas two overlaps.

4. The proposed ensemble learning model forstock selection

This section describes the general architecture of ourproposed model to develop a modular TSK systemwith a weighted ensemble strategy for stock selectionbased on fundamental analysis. In this approach, thesystem uses fundamental data of companies to predictthe future behavior of its stock in the following year andassigns a score to the stocks. Then, the system ranksthe universe of stocks based on the assigned scores.Figure 1 shows the overall framework of the proposedensemble learning model.

This framework starts by collecting fundamentalinformation of companies along with the data prepro-cessing stage to treat the missing variables. We alsoconsider data normalization as one of the necessarydata transformations in forecasting problems [16]. Thisstudy applies the min-max normalization method toobtain a database with all features' values falling in therange of 0 and 1. Due to a large number of fundamentalvariables as the input variables, the most in uentialsubset of variables is selected through stepwise regres-sion analysis. Stepwise regression analysis has beenused successfully for variable selection in stock marketforecasting [69{71,79]. This technique either adds thevariables onward or removes the variables backwardto �nd the best combination of independent variablesfor forecasting the dependent variable. The followingsections describe the other steps of the proposed modelin more detail.

4.1. Data partitioning using the developedhybrid rough-fuzzy noise-rejectionclustering

Due to the variability of fundamental properties of thecompanies within di�erent activity sectors, we believethat data partitioning is a necessary task to have themore homogenous subsets to develop a modular systemfor stock selection. On the other hand, the fundamentalinformation of the companies generally has some out-liers. Therefore, the training data set is partitioned intomultiple overlapping clusters using the proposed rough-fuzzy noise-rejection clustering algorithm, as describedin Section 3. The proposed algorithm has some advan-

Figure 1. The overall framework of the proposed ensemble learning model for stock selection.


tages for our application. First, because of its noise-rejection property, it can handle the outliers withinthe fundamental data set. Second, it bene�ts fromthe fuzzy, possibilistic and rough clustering to resolvethe ambiguity in assigning the objects to the modules.The training patterns of the modules are unique intheir lower approximation, and they are similar to thepatterns of one or more modules in their overlapping orboundary regions. Third, this algorithm represents thecluster members using their rough-fuzzy memberships.The third property is useful for designing an e�cientensemble strategy, as described in Section 4.3.

4.2. Generating TSK fuzzy rule-based systemsfor stock selection

In TSK fuzzy rule-based systems, the knowledge baseincludes multiple fuzzy rules with crisp functions as theconsequent. For the �rst-order TSK system with twoinput variables, the rules are in the form of:

If L1 is FSi1 and L2 is FSi2;

Then yi = ai0 + ai1L1 + ai2L2;

where L1 and L2 are the linguistic variables, FSi1 andFSi2 are their corresponding fuzzy sets, and ai0, ai1,ai2 are the parameters of the system. The systeminferences using crisp reasoning.

The crisp inference of the system is determined asthe weighted average of the individual rule inferencesusing Eq. (17):

y =Pri=1 DOFiyiPri=1 DOFi

; (17)

where r is the number of the rules and DOFi is thedegree of �ring of the ith rule, i.e., the rule's condi-tion memberships aggregated by a t-norm operator asEq. (18):

DOFi = �FSi1(L1) ^ �FSi2(L2); (18)

where ^ is a t-norm operator and �A(x) is the mem-bership degree of x in the fuzzy set A.

In our proposed framework, a unique TSK systemis independently developed for each data partition asan individual learner. Each data partition includes thelower approximation and the boundary of the roughcluster determined by our proposed RFNRC algorithm.The structure and parameter identi�cation of the TSKsystems are described in the following.

We design the TSK rules in a canonical form,where the combination of all the input variables takesplace using the conjunction operator. The structure ofthe TSK rules in the premises is determined by the gridpartitioning. The quality of the TSK system heavilydepends on the partitioning of the input space [80]. We

determine the fuzzy sets for the input variables usingthe modi�ed Adeli-Hung Algorithm (AHA) [71]. Thisalgorithm includes two stages. The �rst stage involvesclustering of the data instances with a topology andweight change neural network, known as Adeli-Hungclustering. The second stage comprises assigning themembership functions to the input space [81]. Thesestages are clearly explained in [82]. In the modi�ed

version of AHA, the input variables are partitionedindividually and the membership functions are deter-mined for individual variables [71]. The modi�ed AHAassigns symmetric triangular fuzzy sets to the inputvariables, where the fuzzy sets of the adjacent labelsoverlap to some extent and their vertex points donot cross. These properties make the system moretransparent [83].

The parameters of the linear conclusion of theTSK rules are determined using the genetic algorithm.Here, GA is used to learn the TSK system because of its exible and powerful search capability [84], especiallyin the case of fuzzy systems learning [85,86]. Thisevolutionary algorithm has been successfully appliedfor both phases of fuzzy modeling, namely structureand parameter identi�cation [80,87]. Among the twocommon approaches for genetic learning of the rule-based systems, i.e., Pittsburgh and Michigan, weapply the Pittsburgh approach to learn the TSK fuzzysystems. In the Pittsburgh approach, one individualencodes the whole rule base of the system. Figure 2shows the encoding scheme of the TSK consequentparameters as a GA chromosome.

The GA chromosomes are evaluated by Informa-tion Coe�cient (IC) as the �tness function. IC is a per-formance measure used for evaluating the forecastingskill of �nancial analysts. It is an appropriate �tnessmeasure for ranking and classifying the stocks in theinvestment universe [60,88]. IC measures the Spearmancorrelation of the ranking that the model assigns to thestocks and their actual rankings in the period shown inBox I, where MRankj;t is the rank of stock j whichis predicted by the model at time t, RRankj;t is theranking of the realized return of stock j at time t, andT is the number of periods in the training set. Ourmodel evaluates the GA chromosomes by the averageIC of the proposed system as Eq. (20):

�tness function: max

(TXt=1

ICt=T

): (20)

Figure 2. Encoding the Takagi-Sugeno-Kang (TSK)consequent parameters as a Genetic Algorithm (GA)chromosome.


ICt =

Psj=1MRankj;t �RRankj;t �

�Psj=1MRankj;t �Ps

j=1RRankj;t�.

nts�Psj=1MRank2

j;t ��Ps

j=1MRankj;t�2�nt��Ps

j=1RRank2j;t �

�Psj=1RRankj;t

�2�nt� ;

t = 1; 2; � � � ; T: (19)

Box I

4.3. Aggregating the TSK systems based onthe proposed weighted ensemble strategy

Our proposed ensemble learning model develops an in-dividual TSK system for each partition of the traininginstances. Similar to the training instances, a new in-stance may belong to multiple partitions with di�erentmembership values. Therefore, our model combinesthe outputs of the corresponding TSK systems usingan ensemble strategy to reach the �nal score of thenew instance. In our proposed modular system, wedesign a new weighted ensemble strategy to combinethe outputs of the TSK systems. In this design, theweight of each module is di�erent for each instance anddepends on the proximity of the instance concerningthe prototype of the module. It uses both roughand possibilistic-fuzzy memberships to calculate therelative importance of the modules as their weights.The ensemble weight of the module i for the instancexj(EWij) is determined by Eq. (21):

EWij =

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

w(uij)w(uij)+

Pkjxj =2AUk

w(ukj)

if xj 2 AUi;w(uij)P

kjxj2BUkw(ukj)+

Pkjxj =2AUk

w(ukj)

if xj 2 BUi;w(uij)P

kjxj2AUkw(ukj)+

Pkjxj2BUk

w(ukj)+P

kjxj =2AUkw(ukj)

if xj =2 AUi

(21)

where w, w, and w correspond to the relativeimportance of the modules regarding the roughpartitions. w is the weight of the module if xj belongsto its lower approximation. w is the weight of themodule if xj belongs to its boundary. In addition,w is the weight of the module if xj does not belongto its partition. Since the instances out of the upperapproximation do not contribute to the moduletraining, w is too small relative to w and w(0 < w �w < w < 1). The relative importance of the modulesdepends on the possibilistic-fuzzy membership of theinstance to its partition (uij), as well.

Figure 3 shows typical partitioning of the dataset in a two-dimensional space. The thickness of

Figure 3. The module's ensemble weights in a typicalrough-fuzzy partitioning.

the arrows corresponds to the ensemble weight of thearrowhead's module. In this �gure, x1 2 AU3 andthe ensemble weight of the third module is very high(EW31 = w(u31)=(w(u31) + w(u11 + u21))). Thisinstance does not belong to the other partitions (x1 =2AU1; x1 =2 AU2); therefore, the weights which areassigned to their corresponding modules are very low(EW11 = w(u11)=(w(u31) + w(u11 + u21));EW21 =w(u21)=(w(u31) + w(u11 + u21))). Another instance,x2, belongs to the boundaries of U1 and U2. Therefore,the ensemble weights of the �rst and second modulesare relatively high based on their possibilistic-fuzzymembership values (EW11 = w(u11)=(w(u11 + u21) +w(u31)), EW21 = (w(u21)=w(u11 + u21) + w(u31))).However, the weight of the third module is too low(EW31 = (w(u31)=w(u11 + u21) + w(u31))).

Consequently, the �nal score of the stock withfundamental properties represented by xj is determinedby Eq. (22):

Scorexj =CXi=1

(EWij � yij); (22)

where yij is the output of module i for the instancexj . Finally, the system selects the stocks according totheir ranked scores at time t.


Table 2. Financial ratios as the possible input variables.

Category Financial ratio

Pro�tability ratios

Percentage of net pro�t to sale, percentage of operating pro�t to sale, percentage of grosspro�t to sale, percentage of gross margin to sale, percentage of net pro�t togross margin,return on assets (after tax), return on equity (after tax), return on working capital, workingcapital return percentage, and �xed assets return percentage

Liquidity ratios Current ratio, quick ratio, liquidity ratio, current assets ratio, and networking capital

Activity ratiosInventory turnover, average payment period, inventory to working capital, current assetsturn over, �xed asset turnover, and total asset turnover

Leverage ratiosDebt coverage ratio, debt to total assets ratio, debt to equity ratio, �xed assets to equityratio, long-term debt to equity ratio, current debt to equity ratio, equity ratio, andinterest coverage ratio

Valuation ratiosActual Earning Per Share (EPS), net dividend per share, price to EPS ratio (P=E), bookvalue per share, dividend yield, price to book ratio (P=B), and capitalization

5. Experimental results

We implemented our proposed ensemble learning modelto develop a modular TSK system for stock selectionamong 150 companies with di�erent activity sectorslisted on TSE. This section describes the collected dataat �rst. Then, it reports the implementation results ofour proposed model step by step. Finally, it analyzesthe performance of the developed modular system andits comparison results.

5.1. DataIn this work, our data comprise the fundamental dataof 150 Iranian companies listed on TSE during 24 �scalyears to develop a stock selection system for investingin this market. These companies are the most liquidcompanies according to six liquidity measures includingthe number of traded shares, the value of traded shares,the number of trading days, the number of trades, theaverage number of shares issued, and the company'svalue in a �scal year. The liquidity measures areaggregated by harmonic mean (Eq. (23)) to �nd themost liquid companies on TSE.

Mj =NPNi=1 Iij

; (23)

where Mj is the jth company's score, N is the numberof indices, and Iij is the value of the ith index forthe jth company. The companies are selected fromthe most liquid companies that were active before2008 from di�erent activity sectors [89]. We selectedthirty-six �nancial ratios as the potential fundamental

variables in �ve categories of pro�tability, activity, liq-uidity, leverage, and valuation ratios [1,3,59,62,65,90].Table 2 reports a list of the selected �nancial ratios. Wecalculated the �nancial ratios of the Iranian companieslisted on TSE using the �nancial statement informationof the companies.

The historical data includes the above-mentioned�nancial ratios and the dividend and split-adjustedclose price for the start and end of the companies'�scal year from March 20, 1991 to March 19, 2014.We used the extending window approach to de�ne thetraining and testing periods. This approach uses allthe available historical data to train the system andthen, applies the trained system in the next periodimmediately after the last training period [91]. Inthis work, we designed �ve series of experiments toevaluate the evolved system for stock selection in �ve�scal years, from March 20, 2009 to March 19, 2014.Table 3 shows the training and the testing periods ofthe �ve series of experiments.

5.2. Implementation of our proposed ensemblelearning model for stock selection

This section reports the implementation results of ourproposed ensemble learning model step by step:

Step 1. This step involves data preprocessing andvariable selection. First, the fundamental data werestudied to handle the missing data and some out-liers. Then, the data were normalized. Finally, themost e�ective �nancial ratios were selected using thestepwise regression analysis. We used the statisticalsoftware, IBM SPSS statistics 20, for setting up


Table 3. The training and testing periods of the �ve experiments.

Experiment series Training period Testing period

Exp 1 Mar. 20, 1991{Mar. 19, 2009 Mar. 20, 2009{Mar. 19, 2010Exp 2 Mar. 20, 1991{Mar. 19, 2010 Mar. 20, 2010{Mar. 19, 2011Exp 3 Mar. 20, 1991{Mar. 19, 2011 Mar. 20, 2011{Mar. 19, 2012Exp 4 Mar. 20, 1991{Mar. 19, 2012 Mar. 20, 2012{Mar. 19, 2013Exp 5 Mar. 20, 1991{Mar. 19, 2013 Mar. 20, 2013{Mar. 19, 2014

the regression forecasting model. We considered thethirty-six mentioned �nancial ratios as the indepen-dent variables and the rate of return as the dependentvariable. We set the probabilities of F statisticsto enter and remove a variable at 0.05 and 0.1,respectively. Due to our experiments, �ve �nancialratios were chosen as the input variables of the TSKsystems. The selected ratios are the Return On Asset(ROA), total asset turnover, equity ratio, dividendyield, and book value per share.

Step 2. We implemented the proposed hybrid rough-fuzzy noise-rejection clustering algorithm to decom-pose the training data set into several overlappingsubsets (i.e., clusters). At �rst, the suitable weightingexponent was selected as m = 2:5, which gave avalue for the trace of the total scatter matrix equalto z=2 (Figure 4). Then, the optimum numberof clusters was identi�ed using the cluster validityindex. According to Figure 5, the rate of reductionin the cluster validity index is very high up to 4clusters, and the index gradually decreases up until 7clusters. We set the number of clusters at C = 4which ensures an almost minimum cluster validityindex as well as su�cient training instances to learnthe individual TSK systems. The noise data arethose with large values of the noise-rejection index(W ), as shown in Figure 6. In our experiments, thethreshold was selected as 2 for calculating the cuto�distance and removing the noise. Then, the FNRC

Figure 4. Selection of the suitable weighting exponent.

Figure 5. Identi�cation of the optimum number ofclusters.

Figure 6. Application of the noise-rejection criterion.

membership degrees were calculated, and the rough-fuzzy data partitions were determined through theiterative procedure of the proposed algorithm. Inthis algorithm, the threshold � represents the sizeof granules of rough-fuzzy clusters. The thresholdcould be determined as the median or the mean of thedi�erence between the highest and the second highestmemberships of all the instances to the speci�edclusters [19,73]. However, when the distribution ofthe membership di�erences is skewed, the medianwould work better than the mean. Therefore, we haveused the following de�nition for � assignment:

� = Medianj=1;2;�� ;N (uij � ukj); (24)


where uij and ukj are the two highest membershipsof instance xj . Based on this de�nition, the thresholdwas set to � = 0:22. Consequently, the modularsystem included four modules with about 600 samplesto train the individual TSK systems.

Step 3. This step resulted in generating the in-dividual TSK systems for each data region, sepa-rately. The structure of the TSK rules' antecedentswas speci�ed through AHA. In our design, threefuzzy sets were allocated to each input variable,interpreted by linguistic labels of low, medium, andhigh. The parameters of the TSK rules' consequentswere learned using GA with Pittsburgh approach.We set the parameters of GA as shown in Table 4.Crossover and mutation rates and reproduction sizeare selected based on some primary experiments. Thepopulation size was selected as 100, showing a betterperformance in multiple runs. Since no signi�cantimprovements were observed after the generationnumber 150, the number of generations was set to200. Four TSK systems were learned for the fourdata partitions on �ve time frames. Table 5 reportsthe performance of the TSK systems in terms of ICon training data subsets. According to this table, theindividual TSK systems could learn a stock selectionsystem with the average IC value of 33.03% in thecase of their corresponding training subsets.

Step 4. This step involved aggregation of the outputsof all of the evolved individual TSK systems to reachan overall stock ranking. The modules' outputs wereaggregated using the proposed weighted ensemblestrategy, and then the stocks were ranked accord-ing to their scores. According to the preliminaryexperiments, we set the w, w, and w to 0.8, 0.6,

Table 4. Parameter settings of genetic algorithm.

Population size 100Number of generations 200Crossover rate 0.7Mutation rate 0.3Reproduction size 20

and 0.2, respectively. The next section reports theperformance of our proposed ensemble learning modelin the �ve testing periods and the comparison results.

5.3. Performance evaluation of our proposedensemble learning model

The performance evaluation of our proposed RFNRC-TSK-MBE (Rough-Fuzzy Noise Rejection Clusteringbased modular TSK system with a Membership-BasedEnsemble) is based on the IC for stock selection onTSE. For comparison purposes, we established threealternative models: single TSK, FNRC-TSK-AveE(FNRC based TSK system with Averaging Ensemble)and RFNRC-TSK-AveE (RFNRC based TSK systemwith Averaging Ensemble). We designed the singleTSK model based on the �rst and third steps of ourproposed RFNRC-TSK-MBE. However, the third stepin the single TSK system uses the whole trainingdataset instead of the training subsets. FNRC-TSK-AveE partitions the dataset using FNRC and, then,assigns the instances to clusters based on the maximummembership. This model develops an individual TSKfor each disjoint subset and applies simple averagingas the ensemble strategy. The RFNRC-TSK-AveEuses the same decomposition and sub-modeling methodwith RFNRC-TSK-MBE, but aggregates the modulesusing the simple averaging ensemble strategy.

Table 6 reports the experimental results of thefour models for stock selection along the �ve testingperiods. The reported numbers are the average resultsof 30 independent runs. According to this table, theRFNRC-TSK-MBE shows a remarkable performancein stock selection. The single TSK system can rankthe TSE stocks with IC of 8.5% on average, which isa good performance. However, our proposed modularsystems could outperform the single system. The �rstmodular system extended by FNRC-TSK-AveE modelcould improve the ranking ability of the TSK systemto IC of 12.74% on average. The ability of the modularsystem is further improved by our proposed clusteringmethod. It reaches 15.24% IC. Above all, our proposedensemble method could boost the predictability of themodular system from IC of 15.24% to 18.15%, onaverage. Similar results are found in all experiments

Table 5. Performance of the individual Takagi-Sugeno-Kang (TSK) systems in their corresponding training data subsetsin terms of Information Coe�cient (IC).

Training subset Exp 1 Exp 2 Exp 3 Exp 4 Exp 5

U1 28.41% 28.11% 28.03% 27.67% 26.67%U2 40.70% 37.84% 30.64% 32.19% 30.45%U3 48.03% 45.45% 43.12% 41.60% 35.11%U4 33.01% 28.04% 31.00% 26.21% 28.35%

Notes: Ui is the ith training subset provided by the rough-fuzzy noiserejection clustering.


Table 6. Performance of our proposed ensemble learning model and the other comparative models in testing periods interms of Information Coe�cient (IC).

Stock selection system Exp 1 Exp 2 Exp 3 Exp 4 Exp 5 Average

Single TSK 6.51% 19.42% 1.81% 14.53% 0.26% 8.50%FNRC-TSK-AveEa 7.51% 15.32% 15.54% 16.48% 8.86% 12.74%RFNRC-TSK-AveEb 11.89% 18.06% 15.17% 17.85% 13.22% 15.24%Our proposed system: RFNRC-TSK-MBEc 12.43% 26.22% 19.86% 18.06% 14.16% 18.15%

a: FNRC-TSK-AveE: Fuzzy Noise Rejection Clustering based TSK system with averaging ensemble;b: RFNRC-TSK-AveE: Rough-Fuzzy Noise Rejection Clustering based TSK system with averaging ensemble;c: RFNRC-TSK-MBE: Rough-Fuzzy Noise Rejection Clustering based modular TSK system with membership-based

ensemble.

Table 7. Results of the student t-test for the pair-wise comparison of the stock selection systems.

Stock selection system RFNRC-TSK-AveE FNRC-TSK-AveE Single TSK

Our proposed system: RFNRC-TSK-MBE 0.0012 [2.91%] 0.0000 [5.41%] 0.0000 [9.64%]RFNRC-TSK-AveE 0.0027 [2.50%] 0.0000 [6.73%]FNRC-TSK-AveE 0.0000 [4.24%]

Notes: The table reports the p-values of tests for the pair-wise dominance of systems' ICs. The di�erence between theIC averages of the respective systems is reported in brackets.

along the �ve periods. The correlation between theRFNRC-TSK-MBE model's output and the one yearforward return is substantially high over each testperiod.

According to the comparison results, the singleTSK model exhibited the weakest performance forstock selection. The reason is di�erent fundamentalcharacteristics of the companies in di�erent activitysectors. Therefore, a modular system works betterfor such a problem. Furthermore, the FNRC-TSK-AveE got a less IC than RFNRC-TSK-AveE, becauseit is di�cult to distinguish a certain boundary betweendi�erent clusters of a data set. Moreover, RFNRC-TSK-MBE arrived at the highest IC value among allthe models. The superiority of the RFNRC-TSK-MBE over the other ensemble models results from itsdecomposition and combination methods. However, itis notable that all the developed TSK systems haveshown good performance in stock selection. As ageneral guideline, 5% is an acceptable IC value ininvestment management [88]. Also, according to aresearch paper published by JP Morgan [92], managerswith ICs between 0.05 and 0.15 can achieve signi�cantrisk-adjusted excess returns. Therefore, TSK systemcan model the stock selection problem properly. Theproposed RFNRC-TSK-MBE model could reach anIC value of 18.15% on average which is much morethan the other stock selection models in the litera-ture [60,88,93,94]. The genetic programming modelprovided by Becker et al. [60] could reach a maximumIC of 8%. Additionally, the average IC of 9% wasobtained in [88] using grammatical evolution. Theauthors claimed that 9% was a high IC value and their

models were successful at the stocks' ranking. Also,Gillam et al. [93] studied the earnings prediction in aglobal stock selection model. They could improve thepredictability of the model to the IC of 6%.

Finally, we carried out the statistical tests to ex-amine whether the proposed model signi�cantly wouldoutperform the other three models or not. The resultsof student t-test are reported in Table 7. Accordingto this table, the RFNRC-TSK-MBE signi�cantly out-performs the other three models at a 99% statisticalsigni�cance level. This table shows that the impactof modularization (FNRC-TSK-AveE over single TSK)is greater than other factors, i.e., the clustering andensemble methods.

In another experiment, the pro�tability of ourproposed model has been investigated for stock clas-si�cation. In our experiment, all the stocks (describedin Subsection 5.1) have been classi�ed into two classes.Similar to [3], we have de�ned Class 1 as the stockswhich elevate the share price to, or more than, 80%within one year. The other stocks have been classi�edas Class 2. In this design, the �rst class constitutes theminority of the data, while it is our interested class.We have applied the over-sampling technique to dealwith the imbalanced data in the training set. In over-sampling, the samples of the rare class are increased bydata replication.

The classi�cation performance of our proposedmodel has been examined in terms of classi�cationaccuracy. For comparison purposes, ANFIS (AdaptiveNeuro-Fuzzy Inference System) has been implementedon the same data base to develop a TSK fuzzy rule-based system for stock classi�cation. The performance


Table 8. Classi�cation performance of our proposed model versus Adaptive Nuro-Fuzzy Inference System (ANFIS) ontesting periods in terms of classi�cation� accuracy.

Model Exp 1 Exp 2 Exp 3 Exp 4 Exp 5 Average

RFNRC-TSK-MBE 77.3% 80.9% 88.1% 79.5% 50.3% 75.22%ANFIS 69.1% 72.6% 72% 61.9% 49.7% 65.06%�: In this experiment, all stocks have been classi�ed in two classes. Class 1 represents

stocks which appreciate in share price equal to or more than 80% within one year.Class 2 contains all other stocks.

Table 9. Performance of our proposed RFNRC-TSK-MBE model in testing periods in terms of appreciation of selectedstocks price.

Exp 1 Exp 2 Exp 3 Exp 4 Exp 5 Average

Average appreciation of the selected stocks 67.2% 89.5% 19.7% 12.2% 128.3% 63.38%Average appreciation of all 150 stocks 55.3% 37.1% 2.2% 8% 117.5% 44.02%Excess appreciation of the selected stocksover all 150 stocks

11.9% 52.4% 17.5% 4.2% 10.8% 19.36%

of ANFIS in stock classi�cation in Dow Jones IndustrialAverage (DJIA) market was previously investigated [3].According to that research, ANFIS outperforms otherneural network models, multi-layer perceptron, andradial basis function in terms of classi�cation accuracyand time complexity. The classi�cation accuracy of ourproposed RFNRC-TSK-MBE model is compared withANFIS model in Table 8. The ANFIS system is trainedin 40 epochs, with two Gaussian membership functionsfor each of the �ve input variables. According to thistable, our proposed model outperforms ANFIS modelin all of the �ve investigated periods. Our proposedmodel reached the average classi�cation accuracy of75.22% in �ve periods, whereas ANFIS could earnclassi�cation accuracy of 65.06%, on average.

Furthermore, the average appreciation of thestock price of the selected stocks (i.e., the stocks thatare classi�ed as Class 1 by the model) is compared withthat of all the investigated stocks in the subsequentyear. Table 9 reports the experimental results in termsof average appreciation. Again, this table con�rms theability of our proposed model in stock selection. TheRFNRC-TSK-MBE could earn the excess appreciationrates of 11.9%, 52.4%, 17.5%, 4.2%, and 10.8% for theselected stocks over all 150 stocks in �ve consecutivetest periods and 19.36% on average.

6. Conclusion

This paper proposed a new ensemble learning model todevelop a modular Takagi-Sugeno-Kang (TSK) systemfor stock selection. The proposed ensemble learningmodel included four stages. The �rst stage involveddata preprocessing and variable selection. The secondstage was about partitioning of the training data intoseveral overlapping regions using the proposed Rough-

Fuzzy Noise Rejection Clustering (RFNRC) algorithm.The proposed algorithm bene�ts from the strengthsof rough, fuzzy and possibilistic clustering, while it issubject to their weaknesses for developing a modularsystem. The diversity of the individual learners wasguaranteed by such a data partitioning algorithm. Atthe third stage, an individual TSK system was gen-erated for each region. The structure and parameteridenti�cation phases of the TSK systems were doneusing Adeli-Hung algorithm and genetic algorithm. Atthe fourth stage, the outputs of the individual TSKsystems were aggregated using the proposed weightedensemble strategy based on the rough-fuzzy member-ships.

In this framework, while handling large data sets,each module might concentrate on knowledge discoverywithin a di�erent region of the problem. Subsequently,all of the modules contributed to problem-solving witha degree based on the similarity of the instance withthe prototypes of the modules. The similarity wasmeasured according to the rough partitions and thepossibilistic-fuzzy memberships.

We implemented our proposed ensemble learningmodel on 150 Iranian companies with di�erent activitysectors listed on Tehran Stock Exchange (TSE) todevelop a modular TSK system for stock selection.Based on the experimental results, our developed mod-ular system could appropriately select the stocks withinformation coe�cient of 18.15% on average, which isa good performance relative to that in the previousresearches [60,88,93]. Furthermore, our proposed sys-tem signi�cantly outperformed the single TSK system(IC=8.5%) and also other modular TSK systems, i.e.,a fuzzy noise rejection clustering based TSK systemwith averaging ensemble (IC=12.74%) as well as aRFNRC based TSK system with averaging ensemble


(IC=15.24%). Upon investigating the comparisonresults, some conclusions can be drawn. First, modularsystems outperformed a single system for problemswith di�erent regions of data characteristics, like stockselection problem, based on fundamental analysis. Sec-ond, the data partitioning using our proposed hybridclustering algorithm led to more capable modularsystems. Additionally, considering the memberships inthe ensemble strategy improved the performance of theensemble model, signi�cantly.

Additionally, the performance of our proposedmodel in stock classi�cation was investigated. Accord-ing to the results, our proposed model outperformedAdaptive Neuro-Fuzzy Inference System (ANFIS) re-garding classi�cation accuracy (75.22% versus 65.06%).Based on this experiment, we can earn much morereturn on investment using the selected stocks byour proposed model for portfolio diversi�cation. Theselected stocks reached 19.36% excess appreciation onaverage, where the average appreciation of all investi-gated stocks was 44.02%.

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Biographies

Somayeh Mousavi received her BSc, MSc, andPhD degrees in Industrial Engineering from AmirkabirUniversity of Technology, Tehran, Iran in 2007, 2009,and 2015, respectively. She is currently an Assis-tant Professor at Industrial Engineering Department,Meybod university, Meybod, Yazd, Iran. Her mainresearch interests include �nancial forecasting, portfo-lio selection, �nancial risk management, fuzzy expertsystems, and applications of arti�cial intelligence andMeta-heuristics in �nancial markets. Dr. Mousavi haspublished her research articles on �nancial decisionmaking using soft computing methods in journals ofExpert Systems with Applications, Knowledge basedsystems, and Applied soft computing.

Akbar Esfahanipour received his BSc degree inIndustrial Engineering from Amirkabir University ofTechnology, Tehran, Iran in 1995. His MSc and PhDdegrees are in Industrial Engineering from TarbiatModares University, Tehran, Iran in 1998 and 2004,respectively. He is currently an Associate Profes-sor at Industrial Engineering Department, AmirkabirUniversity of Technology. He has worked as a se-nior consultant for over 15 years in his specialized�eld of expertise in various industries. His researchinterests are in the areas of forecasting in �nancialmarkets, application of soft computing methods in�nancial decision making, behavioral �nance, �nan-cial resiliency, and analysis of �nancial risks. Dr.Esfahanipour has published his research articles on�nancial decision making in prestigious journals suchas European Journal of Operational Research, Journalof Management Information Systems, Expert Systemswith Applications, Quantitative Finance, Knowledge-


Based Systems, and Applied Soft Computing.

Mohammad Hossein Fazel Zarandi is Professor atthe Department of Industrial Engineering at AmirkabirUniversity of Technology, Tehran, Iran and a memberof the Knowledge-Information Systems Laboratory atUniversity of Toronto, Canada. His main researchinterests focus on big data analytics, arti�cial intelli-gence, data modeling, soft intelligent computing, deeplearning, fuzzy sets and systems, meta-heuristics, andoptimization. Professor Fazel Zarandi has published

over 25 books and Book Chapters, more than 300scienti�c journal papers, more than 200 refereed con-ference papers, and several technical reports in theabove areas, most of which are also accessible onthe web. He has taught several courses in big dataanalytics, data modeling, fuzzy systems engineering,decision support systems, information systems, arti�-cial intelligence and expert systems, systems analysisand design, scheduling, deep learning, simulations, andmulti-agent systems at several universities in Iran andNorth America.

a modular takagi-sugeno-kang (tsk) system based on a modi

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