Two Decades’Research on Decision-theoretic Rough Sets LIU Dun
LI Huaxiong, ZHOU Xianzhong
School of Economics and Management Southwest Jiaotong University Chengdu, China
[email protected] School of Management and Engineering Nanjing University Nanjing, China {huaxiongli, zhouxz}@nju.edu.cn majority rule and just uses one threshold 0.5 to generate the three regions [12]. In contrast to Pawlak model, the requirement of this approach is too loose for real decisions. To overcome these difficulties, probabilistic rough set models are proposed to generalize the 0.5 probabilistic rough sets model, and a pair of threshold parameters is introduced [24,25,35]. Although these models produce the same rough set approximations, only DTRSM consider semantics issues [23]. Three-way decision procedure is introduced into DTRS [21,22,27], two thresholds can be directly and systematically calculated by minimizing the decision cost/loss with Bayes decision theory, which gives a brief semantic explanation with minimum decision risk [26,29,30]. In contrast, the parameters in other probabilistic rough sets model are given by experts with intuitive experience estimating [17,37]. In this paper, we mainly focus on introducing the DTRSM, including the history, development, theories, reduction algorithm and potential applications of DTRS. The rest of the paper is organized as follows. In Section 2, we briefly review the basic concepts of DTRS. The three-way decision procedure in DTRS is illustrated in Section 3. In Section 4, the reduction algorithm in DTRS is presented. Some potential applications of DTRS are introduced in Section 5.
Abstract—The decision-theoretic rough set model (DTRSM) was proposed two decades ago. In this paper, the development of DTRSM, including the theories and the potential applications, are reviewed. With respect to the two semantic issues of DTRSM, three-way decision procedure, attribute reduction methods and potential applications of DTRSM are examined. This paper reviews the history of DTRSM over the last twenty years with a view to its future. Keywords- Decision-theoretic rough sets, Bayesion decision procedure, three-way decisions, probabilistic rough set, semantic explanation
I. INTRODUCTION As an important mathematic methodology to deal with the uncertain problems, rough set theory (RST) uses a pair of approximations to describe a certain set or concept [11]. The equivalence classes induce by equivalence relation are used to construct the approximations. The lower approximation is the union of those equivalence classes that are included in the set and the upper approximation is the union of those that have a non-empty overlap with the set [13]. The two approximations divide the universe into three pair-wise disjoint regions. Positive region is defined by lower approximation, boundary region is defined by the difference between the upper and lower approximations, and negative region is defined by the complement of the upper approximation [21,22]. The Pawlak rough set model does not consider any tolerance of errors. The positive region is those equivalence classes which are completely belong to a finite set, the negative region is the union of those equivalence classes which are completely not belong to a finite set [11]. Instead of the strict requirement of Pawlak rough set, a pair of threshold parameters is used to redefine the lower and upper approximation and probabilistic rough set models [24,25,36]: 0.5 probabilistic rough sets [12], decision-theoretic rough sets (DTRS) [26,29,30], variable precision rough sets (VPRS) [37], rough membership functions [14], parameterized rough set models [10,11], Bayesian rough sets (BRS) [17] and gametheoretic rough set (GTRS) [6] are proposed by allowing certain acceptable level of errors. These models increase our understanding of the RST and its domain of applications. The most challenge of probabilistic rough set models is how to acquire the values of the threshold parameters [5]. The 0.5 probabilistic rough sets model corresponds the simple
II.
A BRIEFLY INTRIDUCTION OF DTRS
Decision-theoretic rough set model (DTRSM) was proposed by Yao and associates in 1990s’ [29,30], and it has been developed for two decades. Before introducing the details of DTRSM, some basic concepts, notations and results of DTRS as well as their extensions are briefly reviewed in this section. [4,6,11-15,17,21-30,35-37] Definition 1: Let U be a finite and nonempty set and R an equivalence relation on U. The equivalence relation R induces a partition of U, denoted by U/R. The pair apr (U , R ) is called an approximation space. For a subset X U / R , the lower approximation and upper approximation of X are defined by: apr ( X ) { x U | [ x ] X } ; apr ( X )
{ x U | [ x ] X z } .
(1)
where [x] is the equivalence class containing x. apr ( X ) is the largest definable set contained in X, and apr ( X ) is the smallest definable set containing X [11]. Based on the rough set
Proc. 9th IEEE Int. Conf. on Cognitive Informatics (ICCI’10) F. Sun, Y. Wang, J. Lu, B. Zhang, W. Kinsner & L.A. Zadeh (Eds.) 978-1-4244-8040-1/10/$26.00 ©2010 IEEE
B
approximations of X, three pair-wise disjoint regions are generated: the positive regions POS( X ) , the boundary regions BND( X ) and the negative regions NEG( X ) : POS( X )
understanding of the levels of tolerance for errors, and the semantic of these model is not clear [23]. Definition 3: Let : { X , X } be a set of 2 states, indicating that an element in X and not in X , respectively. Let $ {aP ,aB ,aN } be a finite set of 3 possible actions, which represent classifying an object in positive region, boundary region or negative region. The losses/costs of those 3 classification actions with respect to different states are given by the 3*2 matrix:
apr ( X ) ,
BND( X )
apr ( X ) apr ( X ) ,
NEG( X )
U apr ( X )
( apr ( X )) .
(2)
For x U , one can make a certainty acceptance decision when xPOS( X ) , a deferment decision when x BND( X ) , and a certainty rejection decision when xNEG( X ) . The three regions lead to three-way decision procedure [21-23]. However, the rules generated by Pawlak rough sets do not allow any tolerance of errors. Soˈprobability is induced to RST and some probabilistic rough set models are proposed to overcome the weakness of Pawlak rough sets [24,25,35].
aP aB aN
| [ x] X | , | [ x] |
(3)
where Pr( X | [ x ]) denotes the conditional probability of the classification, and this simple method for estimating the conditional probability based on the cardinalities of sets is used as an illustration [23]. A main result of DTRSM is to introduce two parameters D and E on Pr( X | [ x ]) to build probabilistic rough sets. If we consider the case with D ! E , the (D , E )- probabilistic lower and upper approximations are defined as follows. (X )
{ x U | Pr( X | [ x ]) t D } ;
apr (D , E ) ( X )
{ x U | Pr( X | [ x ]) ! E } .
apr
(D , E )
{ x U | E Pr( X | [ x ]) D } ,
NEG (D , E ) ( X )
{ x U | Pr( X | [ x ]) d E } .
R ( a P |[ x ])
OPP Pr( X | [ x ]) OPN Pr(X |[ x ]) ,
R ( a B |[ x ])
OBP Pr( X |[ x ]) OBN Pr(X | [ x ]) ,
R ( a N |[ x ])
ONP Pr( X |[ x ]) ONN Pr(X | [ x ]) .
(6)
(P). If R ( aP | [ x ]) d R ( a B | [ x ]) , R ( a P | [ x ]) d R ( a N | [ x ]) , decide xPOS( X ) ; (B). If R ( a B | [ x ]) d R ( a P | [ x ]) , R ( a B | [ x ]) d R ( a N | [ x ]) , decide xBND ( X ) ;
(4)
(N). If R ( a N | [ x ]) d R ( a P | [ x ]) , R ( a N | [ x ]) d R ( a B | [ x ]) , decide x NEG ( X ) . Since Pr( X | [ x ]) Pr(X | [ x ]) 1 , we can simplify the rules (P)-(N) based only on the probabilities Pr( X | [ x ]) and the six losses/costs functions. In addition, by considering the fact the losses/ costs of classifying an object x belonging to X into the positive region POS(X) is less than or equal to the loss of classifying x into the boundary region BND(X), and both of these losses are strictly less than the loss of classifying x into the negative region NEG(X). The reverse order of losses is used for classifying an object not in X. The reverse order of losses/costs is hold for classifying an object x is not in X. Hence, the losses/costs functions may satisfy:
{ x U | Pr( X | [ x ]) t D } ,
BND (D , E ) ( X )
OPN OBN ONN
where, the equivalence class [ x ] of x is viewed as description of x. The Bayesian decision procedure suggests the minimumrisk decision rules:
Similarly, the approximations also divide the universe into three parts as (D , E )- probabilistic positive, boundary and negative regions: POS(D , E ) ( X )
X ( N )
OPP OBP ONP
Where, OPP , OBP and ONP denote the losses/costs incurred for taking action aP , aB and aN when an object belongs to X ; and OPN , OBN and ONN denote the losses/costs incurred for taking action aP , aB and aN when an object does not belong to X . The expected losses/costs associated with taking different actions for objects in [ x ] can be calculated as:
Definition 2: Let S (U , A,V , f ) be an information table. x U , X U , let: Pr( X | [ x ])
X ( P)
(5)
Alternatively, one can define the probabilistic approximations with the three probabilistic regions. Yao et al. introduces Bayesian decision procedure into RST and proposes a decision-theoretic rough set model (DTRSM) [26,29,30], the acceptable level of errors D and E can be automatically computed from losses/cost function, and the optimal decisions with the minimum conditional risk can be directly calculated by using Bayes theory in DTRS [1,26,29,30]. Unfortunately, some probabilistic rough set models as 0.5 probabilistic rough sets model, VPRS and BRS are directly supply the parameters D and E based on the intuitive
OPP d OBP ONP , ONN d OBN OPN .
(7)
With above conditions, (P)-(N) can be rewritten as follows.
B
The formula implies that 0 E d J D d 1 , and the parameter J is no needed. So, the rule (P)-(N) can be rewritten as (P1)(N1), respectively. (P1): If Pr( X | [ x ]) t D , decide xPOS( X ) ; (B1): If E Pr( X | [ x ]) D , decide xBND( X ) ; (N1): If Pr( X | [ x ]) d E , decide x NEG( X ) .
For the rule (P):
R ( a P | [ x ]) d R ( a B | [ x ]) Pr( X | [ x ]) t
(OPN OBN ) . (OPN OBN )(OBP OPP )
R ( a P | [ x ]) d R ( a N | [ x ]) Pr( X | [ x ]) t
(OPN ONN ) . ( OPN ONN ) (ONP OPP )
From the rules (P1)-(N1), we can also get the (D , E )- probabilistic positive, boundary and negative regions as follows.
For the rule (B): R ( a B | [ x ]) d R ( aP | [ x ]) Pr( X | [ x ]) d
POS(D , E ) ( X ) { xU | Pr ( X | [ x ]) t D } ,
(OPN OBN ) . (OPN OBN )(OBP OPP )
BND (D , E ) ( X ) { xU | E Pr( X | [ x ]) D } ,
NEG (D , E ) ( X ) { xU | Pr( X | [ x ]) d E } .
R ( a B | [ x ]) d R ( a N | [ x ])
Although formula (11) produces the same rough set approximations as formula (5), the two threshold parameters in the former formula can be directly calculated from a loss/cost function based on the Bayesian decision procedure, which gives clearly semantic explanations for probabilistic rough sets. However, in probability rough set classification, one may directly use the parameters D and E based on an intuitive understanding of the levels of tolerance for errors, i.e. just adopt the results of DTRS, without a direct reference to a loss function, but they should clearly know that the choice of a particular pair of parameters may be the consequence of losses/costs of various classification decision, whether or not a user is fully aware of the existence of such losses/costs. Actually, the two parameters are closely contact to the loss/cost functions.
(OBN ONN ) Pr( X | [ x ]) t . ( OBN ONN ) (ONP OBP )
For the rule (N): R ( a N | [ x ]) d R ( a P | [ x ]) Pr( X | [ x ]) d
(OPN ONN ) . ( OPN ONN ) (ONP OPP )
R ( a N | [ x ]) d R ( a B | [ x ]) Pr( X | [ x ]) d
(OBN ONN ) . ( OBN ONN ) (ONP OBP )
For simply, we set:
D
E J
(OPN OBN ) , (OPN OBN ) (OBP OPP ) (OBN ONN ) , ( OBN ONN ) (ONP OBP ) (OPN ONN ) . ( OPN ONN ) (ONP OPP )
III.
Furthermore, due to the case with D ! E , we can easily find the following condition on the losses/costs functions: (OPN OBN ) (OBN ONN ) . (9) ! (OPN OBN ) (OBP OPP ) (OBN ONN )(ONP OBP ) b d b bd d , ( a , b, c , d ) ! 0 , ! ! ! a c a a c c
we have: (OPN OBN ) (OPN OBN )(OBP OPP ) !
(OPN OBN )(OBN ONN ) (10) (OPN OBN )(OBP OPP ) (OBN ONN ) (ONP OBP )
!
(OBN ONN ) . (OBN ONN )(ONP OBP )
THE THREE-WAY DECSION PROCEDURE IN DTRS
In Pawlak rough sets, we set the two parameters D 1 and E 0 in formula (11), the two special thresholds divide the universe into three regions, which lead the simple three-way decision rules. The positive rules indicate that an object or object sets for sure belong to one decision class; the boundary rules indicate that an object or object sets partially belong to the decision class; the negative rules indicate that an object or object sets for sure not belong to one decision class [28]. By considering the tolerance of the decision rules, the two parameters D and E which systematically calculate with DTRSM can also divide the universe into three regions, which lead the generalized three-way decision rules. The probabilistic positive rules express that an object or object sets belong to one decision class when the threshold is more than D , which enable us to make decisions of acceptance; the probabilistic boundary rule express that an object or object sets belong to one decision class when the thresholds are between D and E , which lead to the decisions of deferment; the probabilistic negative rules express that an object or object sets not belong to one decision class when the threshold is less than E , which enable us to make decisions of rejection.
(8)
Then, the decision rules (P)-(N) can be expressed as: (P): If Pr( X | [ x ]) t D and Pr( X | [ x ]) t J , xPOS( X ) ; (B): If Pr( X | [ x ]) d D and Pr( X | [ x ]) t E , xBND( X ) ; (N): If Pr( X | [ x ]) d E and Pr( X | [ x ]) d J , x NEG( X ) .
Form the inequality
(11)
For simplicity, we denote the decision procedure as Pawlak three-way model when D 1 and E 0 ; we denote the
decision procedure as two-way model when D E J ; we denote the decision procedure as (D , E ) three-way model when D and E are calculated by DTRSM. Yao has proved that the (D , E ) three-way model is better than Pawlak three-way model and two-way model if it satisfies the formulae (7) and (9), simultaneously [27]. Furthermore, Liu et. al. introduce three-way decision-theoretic rough sets and answer “why” and “how” use DTRSM to solve practical problems [8]. In addition, considered the DTRSM in section 2, Yao suggests using the relative values of the loss function instead of the absolute values in formula (7) [22]. The differences of losses/costs inside pairs of parentheses remain the same; one would obtain the same values for D , E and J , independent of the actual values of the loss function. With the requirement of formula (7), the six loss/cost functions reduce to four basic differences, consisting of O ( P B ) N , O ( B N ) N , O ( N B ) P and O ( B P ) P . The formula (8)
a pair of the same actions of putting two different objects, in C and not in C, respectively, into the positive and boundary regions [22]. To sum up, the ratio in formula (14) focuses on the same status with different actions, and the ratio in formula (15) focuses on the same action with different statuses. The two mathematical transformations give some additional semantic explanations on DTRSM. Slezak uses the odds
Pr( X | [ x ])
Pr( X | [ x ]) t D
Pr( X | [ x ]) D t Pr( X | [ x ]) 1D
Pr([ x ] | X ) Pr( X ) D t Pr([ x ] | X ) Pr( X ) 1D
Pr([ x ] | X ) Pr( X ) D t Pr([ x ] | X ) Pr( X ) 1D
POS(D c, E c ) ( X ) { xU |
(17) Dc
Pr( X | [ x ]) t D c} , Pr( X | [ x ])
BND (D c, E c ) ( X ) { xU | D c d NEG (D c, E c ) ( X ) { xU |
If O ( B N ) N z 0 and O ( N B ) P z 0 , the formula (9) can be
Pr( X | [ x ]) d E c} , Pr( X | [ x ])
Pr( X | [ x ]) d E c} . Pr( X | [ x ])
(18)
However, if we express losses/costs as the six functions of P ( X ) and P (X ) , one can derive the formula (18) from the DTRSM.
rewritten as: (14)
IV.
ATTRIBUTE REDUCTION IN DTRS A reduct in an information table is defined as a minimal subset of attributes that has the same classification power in terms of generalized decision, majority decision, decision distribution, or maximum distribution for all objects in the universe [33]. In general, there are two possible interpretations of the concept of a reduct. The first interpretation views a reduct as a minimal subset of attributes that has the same classification power as the entire set of condition attributes [13]. The second interpretation views a reduct as a minimal subset of attributes that produces positive and boundary decision rules with precision over certain tolerance levels [26,28]. Attribute reduction in the DTRSM is based on these types of probabilistic rules. Reduct construction may be viewed as the search of a minimal subset of attributes that produces positive and boundary decision rules satisfying certain tolerance levels of precision [21,25]. For simplicity, we denote S A and S D as the partitions of the universe U defined by the condition attribute C and decision attribute D, respectively.
The left hand side of inequation (14) concerns the differences of losses for taking two different actions for the same object not in C, and the right hand side considers similar differences of losses for the same object in C. In addition, in the left hand, if we fix on OPN and ONN , the ratio decreases with the increase of OPN . Similarly, if we fix on OPP and O NP in the [22]
.
In the other way, if O ( B N ) N z 0 and O ( B P ) N z 0 , formula (9) can be also rewritten as:
O ( PB) N O ( B N ) N ! O ( BP) P O ( N B) P
(16)
The (D , E ) -probabilistic three region can be rewritten as:
(12)
By interpreting differences of losses in formula (12), one obtains another intuitive interpretation of the required parameters. Let us do some mathematical transformation for formula (9), we have: (13) O ( PB) N O ( N B) P ! O ( B N ) N O ( B N ) N
right hand, the ratio increases with the increase of OBP
Pr( X )Pr([ x ] | X ) . Pr([ x ])
Since Pr( X | [ x ]) Pr( X | [ x ]) 1 , we have:
D
O ( PB) N O ( BP) P ! O ( B N ) N O ( N B) P
to replace the
conditional probability Pr( X | [ x ]) [17]. With the Bayes theorem, Pr([ x ] | X ) can be expressed in terms of the prior probability Pr([ x ] | X ) and the likelihood Pr([ x ] | X ) :
can be rewritten as:
O( PB) N , O ( P B ) N O ( B P ) P O( B N ) N E , O ( B N ) N O ( N B ) P O( PN ) N . J O ( P N ) N O ( N P ) P
Pr( X | [ x ]) Pr( X | [ x ])
(15)
The left hand side of inequation (15) concerns a pair of the same actions, namely, putting an object into the boundary and negative regions, but for two different objects such that one is in C and the other is not in C. On the right hand side considers
Definition 4 [13]: Given an information table S (U , A, V , f ) , an attribute set R C is a Pawlak reduct of C respect to D if it satisfies the following two conditions:
(1). Jointly sufficient condition: POSS R (S D )
parameters questions: z
POSS C (S D ) ;
(2). Individually necessary condition: for any attribute a R , POSS R {a} (S D ) z POSS C (S D ) .
Definition 5 [13]: Given an information table S (U , A,V , f ) , an attribute set R C is a probabilistic reduct of C respect to D if it satisfies the following two conditions: ( D ,E )
( D ,E )
POSS
C
(S D ) ;
(2). For any attribute a R , POSS(D ,E){a} (S D ) z POSS(D ,E ) (S D ) . R
, and we should answer the following two
How do we determine the parameters on a solid theoretical and practical basis? Can we design a systematic method for estimating those parameters?
These questions are clearly answered in section 2, the parameters can be interpreted in terms of loss functions that can be related to more intuitive terms such as losses, costs, and benefits; they can be systematically calculated based on the well known Bayesian decision procedure. The second semantic issue is the interpretation and potential application of probabilistic rules derived from the probabilistic positive, boundary, and negative regions [23], and there still have two related questions.
Then, a natural extension attribute reduction for (D , E )reduct can be also defined as follows.
(1). POSS R (S D )
z
[23]
C
Yao and Zhao point out that definition 5 has some weaknesses such as ignoring of the boundary region, nonmonotocity of probabilistic positive regions, and nonmonotonicity of the quantitative measure [28]. Also, the two parameters D and E are given by experts, and they are lack of semantics. Instead of these opinions, Yao and Zhao give a general definition of a (D , E )- reduct [28,33].
z z
Definition 6: Given an information table S (U , A,V , f ) , Suppose we can evaluate the properties of S by a set of measures E {e1 , e2 ," , en } , an attribute set R C is a reduct of C respect to D if it satisfies the following two conditions: (1). e (S D |S R ) ; e (S D |S C ) for all e E ; (2). For Rc R , e (S D |S Rc ) E e (S D |S C ) for all e E .
In [33], Zhao et. al. explain the Definition 6 again, they state that a reduct R C for positive (boundary) region preservation can be defined by requiring the induced positive (boundary) region is the maximum. In positive region rules, R arg max P C {POS (D , E ) (S D | S P )} .
The first questions can be adequately explained based on the introduced notion of three-way decisions in section 3, and the probabilistic rules with three-way decision procedure has been successful applied in many domains. z Hypothesis testing˖As a sequential test of a statistical hypothesis proposed by Wald, a three-way decision is made, namely, to accept the hypothesis being tested, to reject the hypothesis, and to continue the experiment by making an additional observation [19]. z
Medical clinic˖Pauker and Kassirer use the three-way decision approach to clinical decision making, a pair of a "testing" threshold and a "test-treatment" threshold is used, and the two thresholds divide the actions into three parts: no treatment, need further testing, treatment [10].
z
Products inspecting process: Woodward et al. use three decisions into an inspecting process: accept without further inspection, reject without further inspection, or continue inspecting [20].
z
Documents classification: Li et al. classify documents into three classes based on a three-way decision: relevant documents, irrelevant documents, and possible relevant documents [7].
z
Model selection criteria: Forster considered the importance of model selection criteria with a three-way decision: accept, reject or suspend judgment [2].
z
Environmental management: Goudey discussed threeway statistical inference that supports three possible actions for an environmental manager: act as if there is no problem, act as if there is a problem, or act as if there is not yet sufficient information to allow a decision [3].
z
Data packs selection: Slezak et al. consider a three-way decision when choosing data packs for query optimization, they classify data packs into the relevant, irrelevant, and suspect data packs [18].
z
Email spam filtering: Zhao et al. introduce the DTRSM to classify email, and all the email are divided into three
In non-negative region rules, R arg max P C {NEG(D , E ) (S D | S P )} . In addition, the measure of E can be replaced by the cardinality of the region or the entropy of the region instead of set-theoretic measure of a region [33]. By considering the two parameters D and E can be systematically calculated by the six cost functions, Yao and Zhao use the cost criterion to generate the positive region rules and non-negative region rules with DTRS model [28], and this approach is not only consider the syntax measures such as confidence, coverage and generality, but also consider the semantic measures such as decision-monotocity, cost and risk. V. THE POTENTICAL APPLITIONS ON DTRS Before we study the potential applications in DTRS, Yao suggests there are two semantic issues for DTRSM [23], the first one is the interpretation and estimation of the required
What are the semantic differences between rules from three regions? How to apply probabilistic rules?
categories: spam, no-spam and suspicious [32]. In addition, Zhou et al. also use three-way decision procedure into email filtering, and all the emails are considered as correct email, suspicious email and spam email [34]. z
Oil investment: Yusgiantoro points out there are three classifications for all oilfields: (i) a high potential basin, which is poorly explored and requires exploration and exploitation; (ii) a fair potential basin which contains marginal oil reserves and requires intensive exploration; and (iii) poor potential hydrocarbon, which may or may not contain oil reserves [31]. Furthermore, Macmillan also argues that the oil exploitation decisions should divide into three parts: drill, don't drill, and to continue the decision by acquiring seismic data [9]. To sum up, these successful potential applications with three-way decision procedure illustrate the different semantics in different domains, which explain the second semantic issue for DTRSM.
VI.
CONCLUSIONS
DTRSM has been proposed for twenty years, the histories as well as the development processes of DTRSM are clearly reviewed in this paper. The basic concepts, three-way decision procedure, attribute reduction methods and applications of DTRSM are gradual introduced by surrounding the two semantic issues for DTRSM. This paper gives us a briefly overview of DTRSM, which helps people to study DTRSM more easily. Therefore, as future research, we need to construct some model for DTRS, i.e. extending the two-category classification problems into multiple-category classification problems; extending one agent problems into multiple-agent problems. The semantic studies of DTRS for applications should also be emphasized and enhanced. ACKNOWLEDGMENT The authors thank Professor Yao. Y.Y. for his insightful suggestions. The authors thank the National Science Foundation of China (No. 70971062), the Doctoral Innovation Fund of Southwest Jiaotong University (200907), and the Scientific Research Foundation of Graduate School of Southwest Jiaotong University (2009LD) for their support. REFERENCES [1] Duda, R., Hart, P. Pattern Classification and Scene Analysis. Wiley Press, New York, 1973. [2] Forster, M.: Key concepts in model selection: performance and generalizability. Journal of Mathematical Psychology, 44, 205-231, 2000. [3] Goudey, R.: Do statistical inferences allowing three alternative decision give better feedback for environmentally precautionary decision-making. Journal of Environmental Management, 85, 338-344, 2007. [4] Herbert, J., Yao, J.T.: Criteria for choosing a rough set model. Computer and Mathematics with Application, 57, 908-918, 2009. [5] Herbert, J., Yao, J.T.: Learning optimal parameters in Decision-theoretic rough sets, In: Proceeding of the RSKT2009, LNAI 5589, 610-617, 2009. [6] Herbert, J., Yao, J.T.: Game-Theoretic risk analysis in Decision-theoretic rough sets, In: Proceeding of the RSKT2008, LNAI 5009, 132-139, 2008. [7] Li, Y., Zhang, C., Swan, J. An information fltering model on the Web and its application in JobAgent. Knowledge-Based Systems, 13, 285-296, 2000.
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