Web-Based Medical Decision Support Systems for Three-Way Medical Decision Making With Game-Theoretic...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TFUZZ.2014.2360548, IEEE Transactions on Fuzzy Systems

IEEE TRANSACTION ON FUZZY SYSTEMS 1

Web-based Medical Decision Support Systems forThree-way Medical Decision Making with

Game-theoretic Rough SetsJingTao Yao Senior Member, IEEE, and Nouman Azam

Abstract—The realization of the Web as a common plat-form, medium and interface for supporting human activitieshas attracted many researchers to the study of Web-basedsupport systems (WSS). An important branch of WSS is Web-based decision support systems that provide intelligent supportfor making effective decisions in different domains. We focuson decision making in Web-based medical decision supportsystems (WMDSS). Uncertainty is a critical factor that affectsdecision making and reasoning in the medical field. A three-way decision making approach is an effective and better choiceto lessen the effects of uncertainty. It provides the provisionfor delaying certain or definite decisions in situations that lacksufficient evidence or accurate information in reaching certainconclusions. Particularly, the option of deferment decisions areadded in this approach that provides the flexibility to furtherexamine and investigate the uncertain and doubtful cases. Thegame-theoretic rough set (GTRS) model is a recent developmentin rough sets which can be used to determine the three rough setregions in the probabilistic rough sets framework by determininga pair of thresholds. The three regions are used to obtain three-way decision rules in the form of acceptance, rejection anddeferment rules. In this article, we extend the GTRS modelto analyze uncertainty involved in medical decision making.Experimental results with a GTRS-based approach on differenthealth care datasets suggest that the approach may improve theoverall quality of decision making in the medical field as wellas other fields. It is hoped that the incorporation of a GTRScomponent in WMDSS will enrich and enhance its decisionmaking capabilities.

Index Terms—Game-theoretic rough sets, probabilistic roughsets, Web-based support systems, medical decision making, three-way decision making

I. INTRODUCTION

THE study of Web-based support systems (WSS) aims atdeveloping and transforming existing systems to support

and extend various human activities onto the Web [46], [48].The motivation of WSS research came from the realizationof the Web as a common platform, medium and interfacein supporting and assisting activities like managing, planning,searching and decision making in different fields [45], [51],[62]. An important area of WSS is Web-based decision supportsystems (WDSS) that provide assistance for decision makingin various domains [9], [35]. The WDSS take advantagesfrom opportunities provided by the Web such as increased

This research was partially supported by a Discovery Grant from NSERCCanada and the University of Regina FGSR Dean’s and Verna MartinScholarship programs.

J. T. Yao and N. Azam are with the Department of Computer Science,University of Regina, Regina, Saskatchewan, Canada S4S 0A2, (email: {jtyao,azam200n}@cs.uregina.ca)

availability and effective user interfaces along with advancesin intelligent decision making techniques. Some of the recentexamples of WDSS are found in the fields of group decisionmaking, business, energy and waste management [21], [25],[26], [27], [36], [40]. We consider Web-based medical decisionsupport systems (WMDSS) in this study.

The WMDSS have become a valuable aid for medical prac-titioners in making effective decisions for selecting a course ofaction in medical diagnosis and treatment [8], [42]. Komkhao,Lu and Zhang view the WMDSS from the viewpoint of a rec-ommender system where suitable decision recommendationsare being made by the system [22]. In any case, the WMDSSserve as a platform for integrating evidence-based medicineinto effective and efficient care delivery. There are challengesin designing, developing and deploying WMDSS [37]. Forexample, effective and meaningful user interfaces, recallingand retrieving relevant information, meeting the reliability andaccuracy expectations of decision making and prioritizing andfiltering information to users, just to mention a few [37]. Wefocus on the decision making aspect of WMDSS.

Uncertainty is one of a critical factor that affects reasoningand decision making in the medical field [17], [38]. A three-way decision making strategy that includes a delay, defermentor non-commitment decision option is a better and usefulapproach for reducing the effect of uncertainty in decisionmaking [24], [53], [54], [56]. A particular model in this regardin the medical field was proposed by Pauker and Kassirerfor making treatment decisions [29]. A pair of testing andtest-treatment thresholds were used to define the decisionsof treatment, no treatment and delay treatment based on theprobability of a disease.

The fuzzy and rough set models and their extensions canprovide alternatives for three-way decision making [30], [39],[43], [59], [60], [61]. The shadowed sets which generalizea fuzzy set into a three-valued approximation provides acompelling approach for three-way decision making [32].On the other hand, the rough sets and its extensions alsolead to three-way decisions by interpreting the rough setbased positive, negative and boundary regions as regions ofacceptance, rejection and deferment, respectively. The game-theoretic rough set (GTRS) model is a recent developmentin rough sets that provides yet another approach in thisregard [18], [49]. Particularly, the model provides a thresholdconfiguration mechanism for determining effective thresholdparameters to define the three probabilistic rough set regionsand the implied three-way decisions [3], [5].

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In a recent article, the applicability of GTRS model wasdemonstrated for effectively determining the thresholds bysetting the minimization of overall uncertainty level of theprobabilistic rough set regions as a game objective [5]. Arepeated game was defined that iteratively tunes the thresholdparameters in order to improve the overall uncertainty level ina game-theoretic learning environment.

In this article, we extend these initial results to inves-tigate uncertainty in WMDSS. In Section II we elaboratethe fundamentals of three-way decision making and threeapproaches for three-way decisions, i.e., the GTRS approach,the shadowed sets approach and the threshold approach. Sec-tion III explains an architecture of WMDSS that incorporates aGTRS component for three-way decision making. Section IVdescribes the uncertainties involved in probabilistic roughset regions and a GTRS based approach to minimize theoverall uncertainty of decision regions. Section V providesa demonstrative example for the use of GTRS in obtainingthree-way decisions in medical domain. Finally, Section VIpresents experimental results with the proposed approach ondifferent datasets in the medical domain.

II. THREE-WAY DECISION MAKING

A general framework of three-way decision making wasrecently outlined by Yao in [56].

Considering U as a finite nonempty set of objects and Cas a set of criteria, objectives or constraints. We divide Ubased on C into three disjoint regions, POS, NEG and BNDcalled as positive, negative and boundary regions, respectively.These three regions are utilized to induce rules for three-way decisions. The POS region generates rules for decisionsof acceptance, the NEG region generates rules for rejectionand the BND region generates rules for decision of non-commitment or deferment [53], [54], [56]. The determinationof an object for inclusion in a particular region dependson the degree or level to which it satisfies the criteria inC. Particularly, the POS region consists of objects whosesatisfiability (for criteria in C) is at or above a certain levelof acceptance. The NEG region consists of objects whosesatisfiability is at or below a level of rejection. The BND regionconsists of objects whose satisfiability is above the rejectionlevel but below the acceptance level.

To formally explain the satisfiability of objects and thelevels for acceptance and rejection, Yao introduced the notionsof evaluations of objects and designated values for acceptanceand rejection [56]. It was argued that evaluations providethe degrees of satisfiability and the designated values foracceptance and rejection are the acceptable degrees of satisfi-ability and non satisfiability of objects. A theory of three-waydecisions must consider 1) the construction and interpretationof a set of values for measuring satisfiability and a set ofvalues for measuring non-satisfiability, 2) the constructionand meaning of evaluations and 3) the determination andinterpretation of designated values [56].

There are many approaches to three-way decision making inthe literature [7], [16], [32], [44], [57]. We review only threeof these approaches, namely, the GTRS based approach, theshadowed sets approach and the threshold approach.

A. The Game-theoretic Rough Sets Approach

The probabilistic rough set model has been recognized asa major extension and generalization of the Pawlak rough setmodel [52]. A main result of probabilistic rough sets is thatthe three regions based on a concept C are defined using apair of thresholds (α, β) as,

POS(α,β)(C) = {x ∈ U |P (C|[x]) ≥ α},NEG(α,β)(C) = {x ∈ U |P (C|[x]) ≤ β},BND(α,β)(C) = {x ∈ U |β < P (C|[x]) < α}, (1)

where P (C|[x]) denotes the conditional probability of anobject x to be in C given that the object is in [x] and0 ≤ β < α ≤ 1. Moreover, U is the set of objectscalled universe and the concept C ⊆ U . In the context ofdecision making, the positive, negative and boundary regionsare sometimes interpreted as regions of acceptance, rejectionand deferment decisions, respectively [52]. The probabilisticthresholds (α, β) are interpreted as levels for acceptanceand rejection and the probability P (C|[x]) determines theevaluation of an object given that the object is in [x]. How todetermine the thresholds is a fundamental issue in probabilisticrough sets [55].

The GTRS approach formulates a game to determine therequired threshold parameters. A typical game is defined as atuple {P, S, u}, where: [23]

• P is a finite set of n players, indexed by i.• S = S1× ...×Sn, where Si is a finite set of strategies for

player i. Each vector of the form s = (s1, s2, ..., sn) ∈ Sis called a strategy profile where player i selects strategysi.

• u = (u1, ..., un), where ui : S 7−→ < is a real-valuedutility or payoff function for player i.

For the sake of convenience, the strategy profile with-out the ith player strategy is denoted as s−i =(s1, s2, ..., si−1, si+1, ..., sn). That is s = (s1, s2, ..., sn) =(si, s−i), meaning all the players expect i are committed toplay s−i while player i chooses si.

The outcome of a game is usually determined by using thesolution concept of Nash equilibrium [41]. A strategy profile(s1, s2, ..., sn) is a Nash equilibrium if the following conditionholds,

ui(si, s−i) ≥ ui(s′

i, s−i), where (s′

i ∈ Si ∧ s′

i 6= si). (2)

In other words, si is the best response to s−i for all i. Thisintuitively means that Nash equilibrium is a strategy profilein which none of the players are benefited by changing theirrespective strategies, given the other player’s chosen strategies.

In a GTRS-based game the players are considered in theform of multiple criteria. Each criterion represents a particularaspect of interest such as accuracy or applicability of decisionrules. Suitable measures are selected to evaluate these criteriain the context of rough set based approximation and classi-fication. Each criterion is affected by considering different(α, β) threshold configurations. The strategies are thereforeformulated in terms of changes in probabilistic thresholds [3].The payoff functions represents possible gains, benefits or




performance levels achieved by adapting different modificationof threshold levels. The game is represented using a payofftable which contains the listing of all possible strategiesof every player during a game with their respective payofffunctions or utilities [4].

The use of GTRS is associated with at least two distinguish-ing characteristics compared to other models for three-waydecision making.• Whereas other three-way decision models rely on a

separate and external mechanism to model the deter-mination of thresholds, the GTRS model provides aunified formalism for three-way decisions and thresholdsdetermination [5].

• The ability to determine the threshold levels based on atradeoff solution between multiple criteria can compar-atively lead to more effective and moderate levels foracceptance, rejection and deferment decisions [47].

These properties of GTRS will become more evident inSection V when we consider a demonstrative example.

B. The Shadowed Sets Approach

The shadowed sets introduced by Pedrycz offer a descriptionof a particular fuzzy set by considering three different levelsof membership grades [32]. A shadowed set A is formallyrepresented as,

A : X −→ {0, [0, 1], 1}, (3)

where X represents the universe and the range of A consistsof three components, i.e., 0, 1 and the unit interval [0,1] repre-senting no membership, complete membership and uncertainor partial membership, respectively [33].

The shadowed sets define two operations for an object,namely, the elevation and reduction [33]. The former elevatesthe membership grade of an object to a meaningful andacceptable level of grade and the later reduces the membershipvalue to another meaningful and acceptable level of grade.These two operations play a key role in interpreting andconstructing the three-valued approximation [14]. We elevatethe membership grade A(x) for an object x ∈ X to 1, if A(x)is greater than a certain acceptable level. The membershipgrade is reduced to 0, if A(x) is below than a certain rejectionlevel. The membership grade will be either elevated or reducedif A(x) is between the two levels. An important considerationin this framework is the determination of suitable levels foracceptance and rejection of membership grades. The notion ofthe so called α-cut of a fuzzy set was introduced to define the(α, 1−α) model for interpreting the levels for acceptance andrejection in [33]. A decision-theoretic cost sensitive approachfor determining the two levels was considered in [14]. Itsworth pointing out that further insights may be gained intothe determination and interpretation of the two levels fromgranular computing perspective when they are being consid-ered as levels for admitting or excluding in an informationgranule [31], [50].

The three membership levels or grades in the shadowedsets provide an immediate interpretation from decision makingperspective. For an element x ∈ X with its evaluation

determined by the membership grade represented as A(x). IfA(x) = 1, the object is assumed to be compatible with the setor concept conveyed by A and is accepted for A. If A(x) = 0,the object is not in accordance with the concept determined byA and is rejected for A. If an element receives a membershipgrade in the unit interval [0,1], it is uncertain and we are notable to allocate any numeric grade. A decision of acceptanceor rejection is not being made in this case.

C. The Threshold Approach

The threshold approach was proposed by Pauker and Kas-sirer for making medical treatment decisions [29]. The proba-bility of a disease was used to evaluate a particular decision. Apair of testing and test-treatment thresholds were defined withtesting threshold being less then the test-treatment threshold.The testing threshold defines a level for rejecting or refusing toprovide a treatment. Particularly, if the probability of a diseaseis lesser than the testing threshold, the disease is suggested tobe absent thereby additional diagnosing tests are avoided andtreatment is withheld. The test-treatment threshold defines anacceptable level for providing a treatment. If the probability ofa disease is above the test-treatment threshold, the disease issuggested to be present and immediate treatment is providedwithout any further testing. If the probability is betweenthe two thresholds, the presence or absence of a disease isinconclusive. Additional diagnosing tests are performed witha treatment decision depending on the test outcome.

The two thresholds are affected by different factors such asthe cost, benefit, availability and effectiveness of treatment andthe accuracy, reliability and risks associated with diagnostictesting [10]. The determination of threshold values based onmultiple and sometimes conflicting aspects is a key challengein this context. An approximate rather then exact solution maybe possible that considers a tradeoff among multiple criteriaaffecting the two thresholds. An investigation of some softcomputing technique may be acknowledged for finding andreaching such a solution.

III. AN ARCHITECTURE OF WEB-BASED MEDICALDECISION SUPPORT SYSTEMS

The WMDSS contain various components with functional-ities ranging from supporting end user activities and interac-tion through interfaces, to maintaining and manipulating theknowledge within the system. We adapt the architecture ofrough set based WMDSS proposed by Yao and Herbert [46],[48]. The architecture is slightly modified to incorporate aGTRS component that will assist in decision making. Fig. 1represents the general architecture of WMDSS. We describethis architecture as comprising of an interface, managementand data layers. Each of these layers are now briefly discussed.

A. Interface Layer

The Web and the internet makes up the interface layer. Theclients interact with the system that is designed on the serverside through an interface that is supported by the Web andinternet. The interface is presented to the clients with the




Fig. 1: An Architecture of a Web-based Medical Decision Support Systems

help of Web browsers. It is used by the users to input anyrelevant information that the system require. In addition, itis also responsible for providing services and functionalitieslike searching, storing or obtaining data analysis results forsupporting and making decisions. A carefully designed Webinterface is very critical for the success of the whole system. Ithas to be clear, complete, consistent and should provide userguidance and auto correction facilities.

B. Management Layer

This layer serves as the middleware of the three layerarchitecture. The information from the top or bottom layersare processed at this layer before being presented to upperor lower layers. Intelligent techniques to analyze data such aslogic, inference and reasoning play a key role at this layer.

Some typical components that are required at this levelare Database Management System, Knowledge Management,an intelligent decision making component such as GTRS,Knowledge Discovery/Data Mining and Control Facilities. TheDatabase Management System binds the patient database andthe GTRS component by making the data available from thedatabase to the GTRS component. The Knowledge Manage-ment provides access to the Knowledge Base which containsthe rules database and thresholds database. The GTRS compo-nent makes use of a game-theoretic environment in analyzingdata to acquire knowledge in the form of three-way decisionrules. The Knowledge Discovery component is responsiblefor mining important information based on patients data andknowledge base. In some sense it also serves as an informationretrieval component where it searches and retrieves importantpatterns in data corresponding to different user queries. Finally,the Control Facilities component provides functionalities suchas access rights and permissions to patients data.

C. Data Layer

This layer contains data necessary for the operation of thesystem. A major portion of the data at this layer appears inthe form of patient database which is populated by recordingseries of questions, observations and trails performed onpatients. From decision making perspective, the informationof symptoms corresponding to a disease is the most importantpart of this database. The GTRS and Knowledge Discoverycomponents access this database for analysis and retrievalpurposes, respectively.

The data layer also contains the Knowledge Base componentwhich contains information about three-way decision rulesin the rules database. Rules are learned from the data usingthe GTRS component and are utilized to make decisions ofacceptance, rejection and deferment. The decision rules areaccompanied by the threshold levels in the threshold databasewhich provides the levels for inducing and making three-waydecisions.

IV. ANALYZING UNCERTAINTY IN THREE-WAY DECISIONMAKING WITH GTRS

The GTRS model was recently demonstrated for analyzingthe decision uncertainty involved in the three probabilisticrough set regions [5]. A pair of thresholds controls the uncer-tainties involved in different decision regions [5]. In order todetermine effective threshold values, a game was implementedbetween decision regions with a goal of minimizing the overalluncertainty, resulting in effective and efficient three-way de-cisions [5]. We briefly review the calculation of uncertaintiesinvolved in the probabilistic rough set regions as put forwardby Deng and Yao in [13], [15].

A. Uncertainties in Probabilistic Rough Set Regions

The probabilistic rough set regions are used to create apartition based on a concept C and probabilistic thresholds




(α, β) as [13],

π(α,β) = {POS(α,β)(C),NEG(α,β)(C),BND(α,β)(C)}. (4)

Another partition with respect to concept C is formed as πC ={C,Cc}. The measure of Shannon entropy is used to calculatethe uncertainty in πC with respect to the three probabilisticregions as follows [13]:

H(πC |POS(α,β)(C)) =

−P (C|POS(α,β)(C)) × log P (C|POS(α,β)(C))

−P (Cc|POS(α,β)(C)) × log P (Cc|POS(α,β)(C)), (5)H(πC |NEG(α,β)(C)) =

−P (C|NEG(α,β)(C)) × log P (C|NEG(α,β)(C))

−P (Cc|NEG(α,β)(C)) × log P (Cc|NEG(α,β)(C)),(6)H(πC |BND(α,β)(C)) =

−P (C|BND(α,β)(C)) × log P (C|BND(α,β)(C))

−P (Cc|BND(α,β)(C)) × log P (Cc|BND(α,β)(C)).(7)

Equations (5) - (7) represent the evaluation of uncer-tainty in πC with respect to POS(α,β)(C), NEG(α,β)(C)and BND(α,β)(C) regions. For instance, Equation (5) repre-sents the uncertainty in πC conditioned upon POS(α,β)(C).This is calculated by considering the two sets or compo-nents of the partition πC , i.e., C and Cc and determiningtheir individual uncertainties with respect to POS(α,β)(C),given by, −P (C|POS(α,β)(C)) × log P (C|POS(α,β)(C))and −P (Cc|POS(α,β)(C)) × log P (Cc|POS(α,β)(C)), re-spectively. The conditional probabilities in Equation (5) arecomputed as,

P (C|POS(α,β)(C)) =|C

⋂POS(α,β)(C)|

|POS(α,β)(C)|, (8)

P (Cc|POS(α,β)(C)) =|Cc

⋂POS(α,β)(C)|

|POS(α,β)(C)|. (9)

Conditional probabilities for the other regions are similarlyobtained. When the three regions are interpreted and relatedto three-way decisions, the uncertainty due to probabilisticregions are associated with the uncertainty of their respectivedecisions. For instance, the uncertainty due to positive region,i.e., H(πC |POS(α,β)(C)) is associated with uncertainty in-volved in making the decisions of acceptance. Similarly, theuncertainties H(πC |NEG(α,β)(C)) and H(πC |BND(α,β)(C))are associated with decisions of rejection and deferment,respectively.

The overall uncertainty is calculated as an average uncer-tainty of the three regions. This is also sometimes referred toas conditional entropy of πC given π(α,β) and is determinedas [13],

H(πC |π(α,β)) = P (POS(α,β)(C))H(πC |POS(α,β)(C))

+P (NEG(α,β)(C))H(πC |NEG(α,β)(C))

+P (BND(α,β)(C))H(πC |BND(α,β)(C))(10)

The probability of a particular region, say positive region, is,

P (POS(α,β)(C)) =|POS(α,β)(C)|

|U |. (11)

The probabilities of the other regions are be similarly defined.Equation (10) is reformulated in a more readable form. Con-

sidering the notations of ∆P (α, β), ∆N (α, β) and ∆B(α, β)for representing the uncertainties of positive, negative andboundary regions, respectively, i.e.,

∆P (α, β) = P (POS(α,β)(C))H(πC |POS(α,β)(C)), (12)∆N (α, β) = P (NEG(α,β)(C))H(πC |NEG(α,β)(C)), (13)∆B(α, β) = P (BND(α,β)(C))H(πC |BND(α,β)(C)). (14)

Using Equations (12) - (14), Equation (10) is rewritten as,

∆(α, β) = ∆P (α, β) + ∆N (α, β) + ∆B(α, β). (15)

Equation (15) represents the overall uncertainty with respectto a particular (α, β) threshold pair.

Different probabilistic rough set models represented byvarious (α, β) thresholds will lead to different overall un-certainties. An effective model is obtained by considering asuitable tradeoff among uncertainties of the three regions,controlled by the (α, β) thresholds. We intend to highlightthe role of GTRS model in determining balanced and optimalthresholds that will lead to effective and efficient levels ofuncertainties.

B. A GTRS Approach for Analyzing Uncertainty

We briefly review the GTRS-based approach for analyzingregion uncertainties in this section [5].

The conventional Pawlak model has zero uncertainties inits positive and negative regions while non-zero uncertainty inthe boundary region [13]. The overall uncertainty is howevernot necessarily minimum in this model. To decrease theuncertainty of the boundary, objects must be moved from theboundary to either positive or negative regions. Doing thiswill reduce the uncertainty of the boundary at the cost of anincrease in uncertainty in either positive or negative regions. Insome sense, the boundary region is considered as a commonopponent for both the positive and negative regions. This leadsto the formation of a two-player competitive game where thepositive and negative regions jointly play against the boundaryregion. We will refer to the player representing the positive andnegative regions as immediate decision region, denoted as Iand the player representing the boundary region as deferreddecision region, denoted as D.

The two players in the game are affected by different thresh-old values. Selecting suitable thresholds will allow the playersto compete and guard there personal interests. The strategiesare therefore formulated in terms of different threshold levels.We consider three types of strategies, namely, s1 = α↓(decrease α), s2 = β↑ (increase β) and s3 = α↓β↑ (decreaseα and increase β). This means that the strategy sets for theplayers are the same, i.e., S1 = S2 = {s1, s2, s3}. Althoughthe strategies may be formulated in different ways, the onesconsidered here are based on the initial threshold configurationof (α, β) = (1, 0). An interesting issue in this formulation isto determine how much a threshold based on a certain strategyshould be decreased or increased. This is addressed later in thesame section.




The utilities or payoff functions should reflect possiblebenefits or gains of a particular player. We consider the uncer-tainties associated with different regions to define the payofffunctions. However, an uncertainty value represents a level ofloss or disadvantage measured in the range of 0 to 1 [5]. Anuncertainty of 1 means an extremely undesirable conditionor a minimum possible gain and an uncertainty of 0 meansthe maximum gain. In order to calculate the payoff functionsin terms of gains or benefits for the players, the notion ofcertainty is used which is considered as 1 - uncertainty. Thepayoff functions corresponding to a particular strategy profile,say (si, sj) with associated thresholds (α, β) are defined as,

uI(si, sj) = (CP (α, β) + CN (α, β))/2, (16)uD(si, sj) = CB(α, β), (17)

where,

CP (α, β) = 1−∆P (α, β), (18)CN (α, β) = 1−∆N (α, β), (19)CB(α, β) = 1−∆B(α, β). (20)

The functions uI and uD represent the payoff functionscorresponding to immediate and deferred decision regions, re-spectively, and ∆P (α, β), ∆P (α, β) and ∆P (α, β) are definedas in Equation (12).

Table I represents the game in a payoff table. The rowsrepresent the strategies of player I and the columns representthe strategies of player D. Each cell of the table represents astrategy profile of the form (si,sj), which means that playerI has selected strategy si and player D has selected sj . Thepayoff functions corresponding to a strategy profile (si,sj)is represented as <uI(si, sj),uD(si,sj)>. During the game, aparticular player would prefer strategy si over sj if it providesmore payoff. A strategy profile (si, sj) would be the Nashequilibrium or the game solution if the following conditionsare satisfied for the players,

For I: ∀s′i ∈ S1, uI(si, sj) ≥ uI(s′

i, sj),with(s′

i 6=si),For D: ∀s′j ∈ S2, uD(si, sj) ≥ uD(si, s

′

j),with(s′

j 6=sj).(21)

In other words, none of the players are benefitted by switchingto a different strategy.

We now examine the issue of threshold changes based ona certain strategy during the game. We note from Table I thatthere are four types of fundamental or elementary changesbased on the two thresholds. These changes are defined as,

α− = single player suggests to decrease α, (22)α−− = both the players suggest to decrease α, (23)β+ = single player suggests to increase β, (24)β++ = both the players suggest to increase β. (25)

A threshold pair during a game corresponding to a strat-egy profile, such as, (s1,s1) = (α↓, α↓) is determined as(α−−, β), since both the players intend to decrease thresh-old α (see Equation (23)). In the same way, the profile(s3,s3) = (α↓β↑, α↓β↑) is determined as (α−−, β++) basedon Equations (23) and (25).

Modifying the thresholds repeatedly to improve the utilitylevels of the players will lead to a learning mechanism. Thelearning principle employed in such a mechanism is based onthe relationship between modifications in threshold values andtheir impact on improving the utilities of players (or uncertain-ties of regions). We wish to exploit this relationship to definethe variables (α−, α−−, β+, β++) and to continuously tunethe thresholds in the aim of reaching effective thresholds. Arepeated or iterative game is considered for this purpose.

Consider each iteration of a repeated game that is beingplayed with initial thresholds of (α, β). The game will useequilibrium analysis to determine the output strategy profileand the corresponding threshold pair, say (α

′, β

′). Based on

(α, β) and (α′, β

′), we define the four variables that appeared

in Equations (22) - (25) as,

α− = α− (α× (CB(α′, β

′)− CB(α, β))), (26)

α−− = α− c(α× (CB(α′, β

′)− CB(α, β))), (27)

β++ = β − (β × (CB(α′, β

′)− CB(α, β))), (28)

β++ = β − c(β × (CB(α′, β

′)− CB(α, β))). (29)

The threshold values for the next iteration are updated to(α

′, β

′). The constant c in Equations (27) and (29) is intro-

duced to reflect the desired level of change in thresholds andshould be greater than 1. It is argued that a lower value of callows to fine tune the thresholds based on the data but involvesmore computations [5]. On the other hand, a higher value of cresults in lesser computations however, fine tuning may not bepossible. The iterative process stops when either the deferreddecision region becomes empty, or the positive region sizeexceeds the prior probability of the concept C, or the utilityof deferred decision region exceeds that of immediate decisionregion. These three conditions are mathematically expressedas,

P (BND(α,β)(C)) = 0, (30)P (POS(α,β)(C)) > P (C), (31)uD(α, β) < uI(α, β). (32)

An important consideration in the above framework is theinitial values for the variables α−, α−−, β+ and β++. Onemay set them based on the data itself. If the initial thresholdslead to positive and negative regions that cover majority of theobjects then one would like to slightly modify the thresholds.The values of α− and α−− are set closer to 1.0 and the valueof β+ and β++ are set closer to 0.0. If the initial thresholdslead to positive and negative regions that cover only smallportion of the objects then significant modifications in thethresholds are needed. The values of α− and α−− are setsignificantly away from 1.0 and the values of β+ and β++

are set farther from 0.0. The properties of risks or cost ofmisclassifications can also be considered while setting theinitial values of these variables.

V. AN EXAMPLE OF THREE-WAY TREATMENT DECISIONSWITH GTRS

To illustrate how the GTRS component affects decisionmaking in Web-based medical decision support systems, we




TABLE I: Payoff table for the game

D

s1 = α↓ s2 = β↑ s3 = α↓β↑

I

s1 = α↓ uI(s1,s1),uD(s1,s1) uI(s1,s2),uD(s1,s2) uI(s1,s3),uD(s1,s3)

s2 = β↑ uI(s2,s1),uD(s2,s1) uI(s2,s2),uD(s2,s2) uI(s2,s3),uD(s2,s3)

s3 = α↓β↑ uI(s3,s1),uD(s3,s1) uI(s3,s2),uD(s3,s2) uI(s3,s3),uD(s3,s3)

consider a case of treating a possible disease. The system willmake a treatment decision based on some properties recordedin the form of symptoms or medical tests corresponding to apatient. Table II presents information about previous patientsalong with the past treatments. This information may beretrieved form the patient database. The rows of the tablerepresent the information that is recorded against each patient.The columns S1, S2 and S3 are different characteristics shownby the human body that help in making a treatment decision.They may be of discrete values such as binary if the symptomsare evaluated based on presence or absence. Alternatively, theymay be real numbers reporting some properties like bloodpressure or cholesterol levels. We consider the case of binaryvalues for the sake of simplicity. The last column representssuccessful treatments in the past.

TABLE II: An information table containing patients data

Patient S1 S2 S3 TreatmentP1 1 1 1 YesP2 1 1 0 YesP3 1 0 1 YesP4 0 1 1 YesP5 1 1 0 YesP6 0 1 1 NoP7 0 1 0 YesP8 0 0 1 YesP9 0 1 1 YesP10 1 0 1 YesP11 0 1 1 YesP12 1 0 0 YesP13 1 0 0 NoP14 0 0 0 NoP15 1 0 1 YesP16 1 0 1 YesP17 0 1 0 YesP18 1 0 0 NoP19 0 0 0 NoP20 1 0 1 NoP21 0 1 0 NoP22 0 0 1 NoP23 0 0 1 NoP24 1 0 0 NoP25 0 0 0 NoP26 1 0 0 NoP27 0 1 0 No

In order to do meaningful analysis using a rough sets based

technique, such as, GTRS, we need to have data in the formof an information table. Table II is essentially an informationtable since the set of objects (perceived as patients) is de-scribed by a set of attributes (perceived as set of symptomsand treatment) [58]. Let Xi represents an equivalence classwhich is the set of patients having the same description. Thefollowing equivalence classes are formed based on Table II.

X1 = {P1}, X2 = {P2, P5},X3 = {P3, P10, P15, P16, P20}, X4 = {P4, P6, P9, P11},X5 = {P7, P17, P21, P27}, X6 = {P8, P22, P23},

X7 = {P12, P13, P18, P24, P26}, X8 = {P14, P19, P25}.

The concept of interest in this case is Treatment = Yes. Wewant to approximate this concept in the probabilistic roughsets framework using Equation (1). The association of eachequivalence class Xi with the concept, i.e., P (C|Xi) needsto be determined for this purpose in order to approximate theconcept by three regions. The conditional probability is,

P (C|Xi) = P (Treatment = Yes|Xi) =|Treatment = Yes

⋂Xi|

|Xi|.

(33)

The conditional probabilities of equivalence classesX1, ..., X8 based on Equation (33) are calculated as1.0, 1.0, 0.8, 0.75, 0.5, 0.33, 0.2 and 0.0, respectively.The probability of an equivalence class Xi is determinedas P (Xi) = |Xi|/|U | which means that the probabilityof X1 is |X1|/|U | = 1/27 = 0.037. The probabilities ofother equivalence classes X2, ..., X8 are similarly calculatedas 0.074, 0.1851, 1481, 0.1481, 0.111, 0.1851 and 0.111,respectively.

We now analyze the uncertainties involved in makingdecisions based on the information given in Table II andthe uncertainties involved in probabilistic rough set re-gions discussed in Section IV-A. The uncertainties of theregions based on a certain (α, β) thresholds are calcu-lated by determining the POS(α,β)(C), NEG(α,β)(C) andBND(α,β)(C) regions. Considering the thresholds (α, β) =(1, 0), the three regions based on Equation (1) and condi-tional probabilities P (C|X1), ..., P (C|X8) are POS(1,0)(C) =⋃{X1, X2}, BND(1,0)(C) =

⋃{X3, X4, X5, X6, X7}, and

NEG(1,0)(C) = {X8}. The probability of positive regionis P (POS(α,β)(C)) = |X1

⋃X2|/|U | = 3/27 = 0.111.

The probabilities for the negative and boundary regions aresimilarly obtained.




The conditional probability of C with positive region is,

P (C|POS(1,0)(C)) =

2∑i=1

P (C|Xi) ∗ P (Xi)

2∑i=1

P (Xi)

=1 ∗ 0.037 + 1 ∗ 0.074

0.037 + 0.0740= 1 (34)

The probability P (Cc|POS(1,0)(C)) is computed as 1 −P (C|POS(1,0)(C)) = 1− 1 = 0. The Shannon entropy of thepositive region based on Equation (5) is therefore calculatedas,

H(πC |POS(1,0)(C)) = −1 ∗ log1− (0 ∗ log0) = 0. (35)

The average uncertainty of the positive regionaccording to Equation (12) is ∆P (1, 0) =P (POS(1,0)(C))H(πC |POS(1,0)(C)) = 0. In thesame way, the uncertainty of the negative regionis ∆N (1, 0) = P (NEG(1,0)(C))H(πC |NEG(1,0)(C))= 0, and the uncertainty of the boundary region is∆B(1, 0) = P (BND(1,0)(C))H(πC |BND(1,0)(C)) =0.78. It should be noted that the procedure for calculatinguncertainties will remain the same in case of real or integervalued attributes. This is because individual attribute valuesonly take part in determining equivalence classes. Once theequivalence classes are determined, the uncertainties involvedin different regions are calculated based on the equivalenceclasses contained in different regions. In other words, theattribute values are not directly used in calculating regionuncertainties. The only difference one may expect in case ofreal or integer valued attributes is having more equivalenceclasses due to more variety in the data values.

We highlight the association between thresholds (α, β)and the uncertainties of the probabilistic rough set re-gions. Table III is constructed for this purpose. Eachcell in the table represents three values of the form∆P (α, β),∆B(α, β),∆N (α, β) corresponding to a particularthreshold pair. For instance, the cell at the top left cornerthat corresponds to (α, β) = (1.0, 0.0) has associated regionuncertainties of ∆P (1, 0) = 0.0, ∆B(1, 0) = 0.78 and∆N (1, 0) = 0.0. It is noted that different (α, β) values leadto different region uncertainties. A decrease in α reduces theuncertainty of the boundary region at a cost of an increasein the uncertainty of the positive region. In the same way,an increase in β also reduces the uncertainty of the boundaryat a cost of an increase in the uncertainty of the negativeregion. The (α, β) threshold pair controls the tradeoff amonguncertainties of the three regions. One may expect to minimizethe overall uncertainty by considering a suitable configurationof thresholds. The GTRS model is used for this purpose.

Considering a GTRS-based game outline in Section IV-B.We play the game by considering initial threshold values ofα− = 0.8, α−− = 0.6, β+ = 0.2 and β++ = 0.4. Thepayoff table corresponding to the game is shown in Table IV.The cell in bold, i.e., (0.78,0.72) with corresponding strategypair (β↑, α↓β↑) is the Nash solution of the game defined inEquation (2). The threshold pair corresponding to this strategy

TABLE IV: Payoff table for the example game

D

α↓ β↑ α↓β↑

α↓ 0.855, 0.59 0.84, 0.59 0.775, 0.74

I β↑ 0.84, 0.59 0.86, 0.57 0.78, 0.72

α↓β↑ 0.775, 0.74 0.78, 0.72 0.715, 0.85

profile is (α, β) = (0.8, 0.4). The calculated thresholds withGTRS are used to generate three-way decisions according toEquation (1).

Let us interpret the GTRS results from a medical practi-tioner perspective. Considering a patient x belonging to anequivalence class Xi with conditional probability P (C|Xi),i.e., level of confidence in making a treatment decision. Wehave 78% certainty in making a correct treatment decisionwhen the level of confidence for treating x is set to be greaterthan or equal to 0.8 and the level of confidence for not treatingx is set to be less than or equal to 0.4. When the levelof confidence is less than 0.8 but greater than 0.4, we areunable to decide whether to treat or not, thereby deferring acertain treatment decision. The deferred decisions have 72%certainty which can be further explored and utilized whenadditional information is made available. The threshold levelsaffect the certainty level of the two players. The game-theoreticformulation aims to achieve a suitable tradeoff between thetwo types of decisions.

VI. EXPERIMENTAL RESULTS AND DISCUSSION

We investigate the usage of the GTRS-based approach onfour different health care datasets obtained from the UCImachine learning repository [6]. We briefly describe thesedatasets, namely, contraceptive method choice, haberman’ssurvival, thyroid disease and pima indian diabetes.

A. UCI Datasets

1) Contraceptive Method Choice (CMC): The data in thisdataset is taken from the 1987 National Indonesia Contracep-tive Prevalence Survey. The records corresponds to marriedwomen who were either not pregnant or did not know if theywere pregnant at the time of data collection. The problem isto determine the contraceptive method choice of the womensas No-use, Long-term, or Short-term methods based on demo-graphic and socio-economic characteristics. There are 1,473records and 10 attributes including the class attribute.

2) Haberman’s Survival (HS): This dataset contains casesfrom a study that was conducted between 1958 and 1970at the University of Chicago’s Billings Hospital. The studyconsiders the survival of patients who had undergone surgeryfor breast cancer. The problem is to decide and predict whetherthe patient survived 5 years or longer (≥5 years) or the patientdied within 5 years (<5 years). The dataset contains 306instances and 4 attributes.




TABLE III: Region uncertainties corresponding to different threshold values

β

0.0 0.2 0.4

∆P , ∆B , ∆N ∆P , ∆B , ∆N ∆P , ∆B , ∆N

α

1.0 (0.0,0.78,0.0) (0.0,0.57,0.16) (0.0,0.43,0.28)

0.8 (0.16,0.59,0.0) (0.16,0.41,0.16) (0.16,0.28,0.28)

0.6 (0.29,0.41,0.0) (0.29,0.26,0.16) (0.29,0.15,0.28)

3) Thyroid Disease(TD): There are different datasets inthe UCI repository under this name. We used the new thy-roid dataset. It contains information about five lab tests todetermine a patient thyroid as Euthyroidism, Hypothyroidismor Hyperthyroidism. There are 215 patient records with 6attributes including the class attribute.

4) Pima Indian Diabetes(PID): This dataset contains infor-mation about female patients with at least twenty-one yearsof age at Pima Indian heritage living near Phoenix, Arizona,USA. The problem is to diagnose diabetes as Positive orNegative given a number of physiological measurements andmedical test results. The dataset consists of 768 records with9 attributes.

B. Experimental Results with GTRS-based Method

In order to apply the GTRS-based repetitive thresholdmethod, we started with initial values for the variables definedin Equation (27) as α− = 0.9, α−− = 0.8, β+ = 0.1and β++ = 0.2. The next values of these variables wereautomatically obtained as mentioned in Section IV-B. Theinitial thresholds were set to (α, β) = (1, 0) in all experimentsfor starting the method.

TABLE V: Dataset CMC, category = No-use

Region Size Thresholds CertaintyPOS BND NEG α β uI uD

19.3 57.9 22.7 1.000 0.000 1.000 0.43722.0 53.0 25.0 0.800 0.100 0.975 0.48724.2 50.2 25.6 0.720 0.120 0.959 0.51724.6 49.8 25.6 0.676 0.127 0.956 0.52324.6 49.8 25.6 0.668 0.129 0.956 0.52327.7 46.7 25.6 0.653 0.130 0.934 0.55928.2 43.2 28.5 0.606 0.149 0.918 0.58831.2 40.3 28.5 0.536 0.149 0.896 0.62031.2 38.7 30.1 0.501 0.168 0.889 0.63342.4 27.4 30.1 0.476 0.173 0.809 0.754

Table V shows the results for category No-use of CMCdataset. Each row of the table represents a single iterationof the repetitive game. The first row has the initial thresholdsettings of (1,0) which corresponds to the Pawlak model. Wehave maximum utility for player I , i.e., 1.0 and moderate

TABLE VI: Dataset CMC, category = Long-term


4.5 45.8 49.8 1.000 0.000 1.000 0.5574.9 44.5 50.6 0.800 0.100 0.994 0.5684.9 44.5 50.6 0.782 0.102 0.994 0.5684.9 44.5 50.6 0.764 0.105 0.994 0.5684.9 44.5 50.6 0.745 0.107 0.994 0.5684.9 44.5 50.6 0.727 0.109 0.994 0.5684.9 43.3 51.8 0.709 0.114 0.988 0.5785.6 42.6 51.8 0.681 0.114 0.983 0.5865.6 42.6 51.8 0.671 0.115 0.983 0.5869.0 39.2 51.8 0.649 0.117 0.962 0.625

10.7 35.8 53.4 0.598 0.136 0.944 0.65714.6 30.5 54.9 0.560 0.153 0.916 0.711

TABLE VII: Dataset CMC, category = Short-term


10.5 58.0 31.4 1.000 0.000 1.000 0.43112.8 55.1 32.1 0.800 0.100 0.985 0.46414.7 53.2 32.1 0.696 0.107 0.974 0.48618.3 49.6 32.1 0.633 0.111 0.950 0.52920.6 46.2 33.2 0.579 0.130 0.930 0.56323.2 43.6 33.2 0.501 0.139 0.913 0.592

utility for player D, i.e., 0.437. As the method continues,the thresholds are modified based on game-theoretic analysisand game outcomes. The probabilistic positive, negative andboundary regions change in size based on configuration ofthresholds which ultimately leads to different uncertaintylevels and utilities for players. The size of positive and negativeregions keeps on increasing while the boundary region keepson decreasing. The method stops at the ninth iteration as oneof the stop conditions described in Equations (30) - (32) wasreached. The certainty of definite decision region decreased by19.1% while the certainty of deferred decision region increasedby 31.7%. The definite decision region increased by 30.5%which means that we have more immediate decisions. An




TABLE VIII: Dataset HS, category = ≥ 5 years


7.8 92.2 0.0 1.000 0.000 1.000 0.20320.6 79.4 0.0 0.900 0.200 0.972 0.28149.0 48.4 2.6 0.759 0.263 0.860 0.535

TABLE IX: Dataset HS, category = < 5 years


0.0 92.2 7.8 1.000 0.000 1.000 0.2030.0 79.4 20.6 0.900 0.100 0.972 0.2810.0 73.2 26.8 0.759 0.131 0.956 0.3242.6 65.7 31.7 0.695 0.154 0.930 0.3949.8 54.2 35.9 0.570 0.175 0.882 0.513

intuitive interpretation of these results is that by decreasingour expectation of making 100% certain decisions to 80.9%certain decisions, we included 30.5% of additional cases toour definite or immediate decision region that were previ-ously unknown to us. The final threshold configuration is(α, β) = (0.476, 0.168).

Table VI shows the results for category Long-term of CMCdataset. The method reaches the stop criteria in slightly moreiterations. The configured thresholds in this case correspond to(α, β) = (0.56, 0.154). The certainty of the deferred decisionregion increased by 15.4% at a cost of 8.4% decrease in thecertainty of definite decision region. Table VII presents theresults for the category short-term of the same dataset.

Tables VIII and IX shows the results for the categories ofHS dataset. The deferred decision regions of the two categoriesare successfully increased in a few iterations. For both thecategories the certainty increases of deferred decision regionsare much more and greater than the certainty decreases ofimmediate decision regions. This is acceptable as the overallcertainty or utility of the two players has improved. It seemsrelevant in this case to investigate a suitable value of constantc (used in Equation (27)) to fine tune the thresholds and toachieve and improve the utilities of the players.

The results obtained with the dataset TD and PID aresummarized and presented together in Table X. The resultsare quite similar to those obtained with the datasets CMC andHS.

Fig. 2 summarizes the changes in immediate decision regionsize and the certainties of the two players. A set of three barsare shown for each category of the considered datasets. Theblack bars correspond to the percentage increases in the sizeof immediate decision regions. The white bars represent theincreases in certainty of immediate decision region, i.e., uI .The grey bars represent the decreases in certainty of deferreddecision regions, i.e., uD. Considering the CMC dataset, theresults are quite encouraging for the three categories. The sizesof definite decision regions increased by 30.5%, 15.3% and

14.4% for the three categories as indicated by the blue bars.Considering the green and brown bars for the three categoriesof this dataset. The increases in certainty of deferred decisionregions are more than decreases in certainty of the immediatedecision regions for the three categories. This means that theoverall uncertainty is improved to a certain extend. The HSdataset showed the most significant improvements. The bluebars for the two categories of this dataset indicate that theimmediate decision regions increased by 43.8% and 38% forthe two categories. A considerable improvement in certaintyof deferred decision regions are also noticed from the brownbars indicated in the figure. The table also suggests someimprovements, although not very significant on the TD andPID datasets.

C. Further Analysis and Disscussion

We further analyze the decision making capabilities ofGTRS by considering the quality of obtained decision regions.The property of classification accuracy of decision regions isconsidered for this purpose which is defined as [5],

Accuracy(α, β) =Correctly classified by POS(α,β) and NEG(α,β)

Total classified by POS(α,β) and NEG(α,β)

.

(36)The option of deferred decision in GTRS reduces somemisclassifications which will lead to an improved accuracycompared to other methods. In order to have a better insightand comparable results, we also include the deferred decisionrate, i.e., the percentage of decisions that are being deferred(also called as non-commitment rate [15]). The deferred ornon-commitment rate is the relative size of the boundaryregion with respect to the universe, i.e.,

non−commitment(α, β) =|BND(α,β)(C)|

|U |. (37)

Let us first consider the results for the CMC dataset. TheGTRS-based approach was used to learn the thresholds whichwere used to determine the decision regions and the associateddecision accuracy according to Equation (36). A decisionaccuracy of 90.57% was noted with non-commitment rateof 34.3%. This means that the GTRS was able to classify65.7% with a success or accuracy rate of 90.57%. Additionalinformation is needed to make certain decisions about theremaining 34.3% of the objects.

An important issue while considering the GTRS basedperformance is that it requires additional information for someobjects. In the absence of such information, one may feeluncertain with remaining doubts about its performance. Letus look at the GTRS performance in a worst case scenariowhere we have no access to additional information and weare asked to make certain decisions about the deferred cases.Suppose that random decisions are being made with a 50%chance of being correct and another 50% of being incorrect.In this case, the GTRS provides 90.57% correct decisionsfor 65.7% of the objects and 50% correct decisions for theremaining 34.3% of the objects. The average accuracy in thiscase is (90.57 × 0.657) + (50 × 0.34) = 76.65%. This iscomparable and to certain extend quite encouraging when we




TABLE X: Results for datasets TD and PID

Dataset Category Iterations Config.Region Size Thresholds Certainty

POS BND NEG α β uI uD

TD

Euthy. 6Intial 58.6 22.8 18.6 1.000 0.000 1.000 0.772Final 64.7 11.2 24.2 0.636 0.278 0.935 0.889

Hypothy. 16Intial 7.0 14.9 78.1 1.000 0.000 1.000 0.858Final 14.9 7.0 78.1 0.404 0.000 0.967 0.932

Hyperthy. 9Intial 11.6 7.9 80.5 1.000 0.000 1.000 0.931Final 11.6 5.1 83.3 0.502 0.275 0.979 0.952

PIDNegative 14

Intial 59.5 12.0 28.5 1.000 0.000 1.000 0.881Final 61.1 9.2 29.7 0.503 0.264 0.968 0.908

Positive 23Intial 28.5 12.0 59.5 1.000 0.000 1.000 0.881Final 32.2 6.8 61.1 0.501 0.431 0.942 0.932

Fig. 2: Summary of changes in certainty and the size of immediate decision region

consider some of the previous results for the same dataset. Forinstance, a fuzzy decision tree approach was reported to have76.2% accuracy [12]. The same research reported accuraciesof 63.86%, 50.8% and 40.85% for the neural network, supportvector machine and k-nearest neighbour approaches. Someother known approaches like C4.5 and k-means were reportedto have 61.67% and 66.66% accuracies, receptively [1].

Another important characteristic of the GTRS is that itrequires additional information about a proportion of theobjects in order to improve its performance. For example,in case of the CMC dataset, 34.3% of the objects needsadditional information to be certainly classified. There is noexplicit mechanism in conventional two-way decision methods

to identify the cases that require additional investigations. Theyrequire additional information for all objects in order to furtherimprove their performance.

We now consider the case of the HS dataset. The GTRS-based method achieved a decision accuracy of 83.5% with anon-commitment rate of 42.9%. The deferred decision regionin this case is slightly larger compared to the CMC datasetwhere it was found to be 34.3%. The size of this region isexpected to decrease if we lower the expectation of beinghighly accurate or when additional information is being madeavailable. Once again, we consider the constraints of noadditional information and forced two-way certain decisions.Under these constraints, the GTRS makes 83.5% correct deci-




sions for 57.1% objects and a 50% chance of correct decisionsfor the remaining 42.9% objects. The average accuracy inthis case is 69.12% which is worst but still comparable tosome other approaches in the literature. For instance, C4.5 wasreported to have 71.7% accuracy [11]. The Naive Bayesain andSVM were reported to have accuracies of 73.83% and 73.52%,respectively [28].

The results for the datasets TD and PID were quite encour-aging. The decision accuracies for the TD and PID datasetswere 98.64% and 98.66% with a non-commitment rates of7.7%, 8.1%, respectively. The deferred decision regions forthese datasets were comparatively small implying many certaindecisions. This means that correct decisions are confidentlymade for majority of the objects. If we consider the constraintsas discussed above, i.e., no additional information and forcedtwo-way certain decisions, the average accuracies are 94.89%and 94.72% for the TD and PID datasets, respectively. Theseresults are quite promising when compared with some ofthe existing results. For instance, an information gain basedapproach and Fuzzy C-Means were reported to have accuraciesof 95.9% and 83.7%, respectively, for the TD dataset [2],[20]. On the other hand, an approach based on combiningregression with neural networks and another approach basedon dicriminant analysis with support vector machine demon-strated accuracies of 80.21% and 79.16%, respectively, forthe TD dataset [19], [34]. Further improvements in the GTRSperformance is expected by providing additional informationfor the remaining 7.7% and 8.1% of the objects in the deferreddecision regions of the TD and PID datasets, respectively.

The results presented in this section advocate for the useof a GTRS component in WMDSS as an alternative wayfor obtaining effective decision support for different medicaldecision making problems.

VII. CONCLUSION

We examine and extend the decision making capabilitiesof Web-based medical decision support systems. Decisionmaking in the medical field is typically associated with a levelof uncertainty. A three-way decision making approach is auseful and better choice to reduce the impact of uncertainty inmedical decision making. The concept of three-way decisionswas recently incorporated in the rough set theory for inter-preting the three rough set regions as regions of acceptance,rejection and deferment. We consider three-way decisions inthe probabilistic rough set model which is the generalizedrepresentation of rough sets. A fundamental problem in thismodel is the interpretation and determination of decisionthresholds that define the division between the three roughset regions. The game-theoretic rough set or GTRS modelhas been demonstrated for determining effective thresholdsfor probabilistic rough sets by analyzing the uncertaintiesinvolved in the probabilistic rough set regions and the impliedthree-way decisions. We incorporated a GTRS component ina WMDSS architecture in the aim of enriching its decisionmaking capabilities. The benefits from a GTRS componentis demonstrated by considering a GTRS-based method forimproving the overall uncertainty level of the three-way clas-sification. The method iteratively modifies the thresholds in

order to improve the overall uncertainty of rough set basedregions and their respective decisions. The method providesencouraging results on four different medical datasets.

The uncertainty analysis of decision making sets up themotivation for analyzing different decision making aspectswith GTRS, such as, the risks, errors, costs and benefits as-sociated with medical decision making. Different competitiveor cooperative games may be setup to determine cost effectiveand balanced thresholds based on these aspects.

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JingTao Yao (M’01-SM’05) received the Ph.D.degree from the National University of Singapore,Singapore.

He is currently a Professor with the Departmentof Computer Science, University of Regina, Regina,SK, Canada. He taught in the Department of In-formation Systems, Massey University, PalmerstonNorth, New Zealand; the Department of InformationSystems, National University of Singapore, Singa-pore; and the Department of Computer Scienceand Engineering, Xian Jiaotong University, Xian,

China. He has edited many volumes of conference proceedings. Recently,he edited books: Novel Developments in Granular Computing: Applicationsfor Advanced Human Reasoning and Soft Computation (Hershey, PA: IGIGlobal, 2010) and Web-based Support Systems (New York: Springer, 2010).His research interests include granular computing, rough sets, data mining,electronic commerce, Web intelligence, and Web-based support systems. Hehas over 100 papers in these areas.

Dr. Yao serves as a member of Editorial Boards of the International Journalof Data Mining, Modelling and Management, the International Journal ofGranular Computing, Rough Sets and Intelligent Systems, the Journal ofEmerging Technologies in Web Intelligence, and the Journal of IntelligentSystems. He also serves as the Guest Editor of Neurocomputing, the Interna-tional Journal of Approximate Reasoning, the International Journal on Multi-Sensor, Multi-Source Information Fusion, the Transactions on Rough Sets,and Fundamenta Informaticae. He was Elected Chair of Steering Committeeof International Rough Set Society (2012-2014). He has been a Chair or amember of Program Committee of numerous international conferences.

Nouman Azam is a Ph.D, Candidate in the Depart-ment of Computer Science, University of Regina,Canada. Before arriving in Canada, he completed hisM.Sc. in Computer Software Engineering from Na-tional University of Sciences and Technology, Pak-istan and B.Sc. in Computer Science from NationalUniversity of Computer and Emerging Sciences,Pakistan in 2007 and 2005 respectively. His researchinterests include rough sets, game theory, three-way decision making, decision support systems andmultiple criteria decision analysis.

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