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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: DOI 10.1109/TFUZZ.2019.2953024, IEEETransactions on Fuzzy Systems

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. XX, NO. X, MONTH YEAR 1

Fusing Fuzzy Monotonic Decision TreesJieting Wang, Student Member, IEEE, Yuhua Qian∗, Member, IEEE, Feijiang Li, Student Member, IEEE,

Jiye Liang and Weiping Ding, Senior Member, IEEE

Abstract—Ordinal classification is an important classificationtask, in which there exists a monotonic constraint betweenfeatures and the decision class. In this paper, we aim at developinga method of fusing ordinal decision trees with fuzzy roughset based attribute reduction. Most of the existing attributereduction methods for ordinal decision tables are based on thedominance rough set theory or significance measures. However,the crisp dominance relation is difficult in making full useof the information of attribute values; and the reducts basedon significance measures are poor in interpretability and maycontain unnecessary attributes. In this paper, we firstly definea discernibility matrix with fuzzy dominance rough set. Withthis discernibility matrix, multiple reducts can be found, whichprovide multiple complementary feature subspaces with originalinformation. Then diverse ordinal trees can be established fromthese feature subspaces, and finally, the trees are fused bymajority voting. The experimental results show that the proposedfusion method performs significantly better than other fusionmethods using dominance rough set or significance measures.

Index Terms—Ordinal classification, ensemble learning, at-tribute reduction, fuzzy dominance rough set, discernibilitymatrix

I. INTRODUCTION

FUZZY rough set theory is known as a powerful model foranalyzing uncertainty in big data [1]–[5]. In the rough

case, a crisp set is provided with a lower approximationand an upper approximation, which allow for a granularrepresentation of knowledge and an excellent description ofthe uncertain region. In the fuzzy case, relations betweenobjects and sets or relations among objects are characterizedby degrees of membership. This allows for great flexibility indealing with imprecise information. Fuzzy rough set theorycombines the advantages of fuzzy sets and rough sets. It canhandle uncertainty in nominal or real-valued attributes and hasbeen successfully applied to machine learning, logical rea-soning, pattern recognition, intelligent information processing,and other fields [6]–[10].

An important achievement of rough set theory is that ofattribute reduction in the decision table. Attribute reductionremoves the redundant attributes and preserves the necessaryattributes that can maintain the same discrimination informa-tion as the original decision table. Great progress has madeon attribute reduction in the past few decades. Slezak [11]

∗: The corresponding author.J.T. Wang, Y.H. Qian and F.J. Li are with the Institute of Big

Data Science and Industry, Shanxi University, Taiyuan 030006, ShanxiProvince, China; Y.H. Qian and Jiye Liang are with the Key Lab-oratory of Computational Intelligence and Chinese Information Pro-cessing of Ministry of Education, Shanxi University, Taiyuan 030006,Shanxi Province, China; Weiping Ding is with the School of Infor-mation Science and Technology, Nantong University, Nantong 226019,China. (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected].)

reviewed the attribute reduction methods that kept the entropyinvariant. Tsang et al. [12] developed a granularity and re-duction theory of fuzzy rough sets. Chen et al. [13] proposeda reduction method based on fuzzy rough sets with Gaussiankernel. Wang et al. [14] proposed a task fitting selection modelwith fuzzy rough sets to guarantee that the membership degreeof a sample reaches the maximal value in its class. Ding andLin et al. proposed a series of efficient reduction methods forlarge scale databases with fuzzy rough sets [15]–[17].

Ordinal classification problems widely exist in practicalscenarios such as credit risk assessment, product performanceevaluation, university ranking and so on [18], [19]. In ordinaldecision tables, the values of attributes are characterized bypreference relationships. In addition, there is a monotonicrelationship between the feature attribute and the decisionattribute. Generally, the objects with better feature valuesshould not belong to a worse decision class, such as thestudents with higher scores should have higher grades. Saiand Yao et al. [20] gave the precise definition of generalordinal tables and proposed a framework to transform theordinal table into a traditional one. Another effective approachto solve ordinal classification is the dominance rough set basedmethods [21]. Dominance rough set replaces the equivalencerelation in the classical rough set with the dominance relations.Based on dominance rough set, Hu et al. [7] defined themonotonic rank information entropy to measure the quality ofattributes in the ordinal decision tables, and then proposed amonotonic decision tree REMT to solve ordinal classificationproblem. Just as CART for traditional classification, REMTplays an important role in obtaining monotone consistent,noise robust and highly interpretable rules.

To further improve the performance of REMT, Qian et al.[8] and Wang et al. [22] employed ensemble learning method.Ensemble learning is one of the most promising methods toimprove the generalization performance of learning systems,especially for decision tree systems [23]–[30]. The main ideaof ensemble learning is to fuse multiple different classifierswith weak performance. The diversity and accuracy of the baseclassifiers are two crucial factors in determining the ensembleperformance. To guarantee the performance of ensemble, Qianet al. [8] and Wang et al. [22] employed attribute reductionbased on dominance rough set to generate multiple featuresubsets, and then constructed multiple monotonic decisiontrees on these subsets. The fusion of the monotonic treesperformances better than the individual one. This inspires usthat the strategy of attribute reduction can be well appliedto ensemble learning. In this paper, we aim to use ensemblelearning with attribute reduction technique to solve the ordinalclassification problem.

Most of the existing researches for ordinal attribute re-




duction are based on the dominance rough set theory. Qianet al. [8] defined a variable monotonic dependency measurebased on dominance relation to conduct attribute reduction.Wang et al. [22] defined a discernibility matrix preserving themonotonic consistency based on dominance rough set theory.However, the dominance rough set is an extension of theclassical rough set theory. It inherits the limitation of the roughset theory: poor ability to deal with real-valued attributes, suchas having a bias to the attributes with more values. Fuzzydominance rough set theory uses fuzzy dominance relationinstead of crisp dominance relation to measure the relationshipbetween objects. Fuzzy dominance relation can not only geta preference relation between attribute values but also get apreference degree [31]. In the area of fuzzy rough set, Huet al. [32] defined three significance measures and then usedthe forward heuristic algorithm for ordinal attribute reduction.However, a reduction method based on the heuristic algorithmsmay contain unnecessary attributes [33]. Besides, the heuristicalgorithms usually get one reduct and the result is easilyaffected by the reading order of attributes.

In addition to the reduction methods based on the signifi-cance measures, Skowron and Rauszer put forward a reduc-tion method with a discernibility matrix and a discernibilityfunction [34]–[36]. Each element of the discernibility matrixstores an attribute subset which distinguishes a pair of objectsin some specified sense. The discernibility function performsconjunction and disjunction operation on the elements of thediscernibility matrix to get multiple attribute subsets. Thereduction result based on the discernibility function is usuallycalled as a complete reduct because this kind of result con-tains all possible attribute subsets defined in the discernibilitymatrix. In addition, the attribute subsets generated by thediscernibility matrix maintain the discrimination informationof the original attribute set from different dimensions, whichprovides a good foundation for establishing multiple classifierswith certain accuracy and diversity. Thus, the result of attributereduction based on a discernibility matrix is more suitable tothe requirements of ensemble learning than that based on thesignificance measures.

According to the above analysis, we focus on using en-semble learning with discernibility-matrix-based attribute re-duction technique to improve the performance of ordinalclassifiers. Particularly, we firstly propose the definition ofupward and downward local positive region of the ordinaldecision table. Based on the definition of lower approximationin [12], we propose a discernibility matrix that preserves theupward local positive region invariant. Then, based on thediscernibility matrix, multiple reducts can be obtained. Finally,each attribute subset is used to construct a fuzzy rank entropymonotonic decision tree, and all of them are fused to obtainthe final predicted results.

The rest of the paper is organized as follows. In Section II,we introduce the definitions of rough set theory and presentthe existing methods of attribute reduction for ordinal decisiontable. In Section III, we give the definition of discernibilitymatrix preserving the upward local positive region invariant.We also present the algorithms for fusing fuzzy monotonicdecision trees in this section. The experimental results and

analyses are presented in Section IV. In the end, we concludeand put forward the future work in Section V.

II. PRELIMINARIES

In this section, we give the fundamental definitions indominance rough set theory and introduce some discernibilitymatrixes and some significance measures, which were pro-posed for attribute reduction for ordinal decision table.

A. Dominance Based Rough Sets

Let DT = (U,A ∪ {d}) be an ordinal decision table,where U = {x1, x2, · · · , xn} is a set of objects, A ={a1, a2, · · · , am} is a set of feature attributes and d is thedecision class with values {d1, d2, ..., dK}. Let v(x, a) be thevalue of x with respect to the attribute a ∈ A, we considerthat v(x, a) ∈ Z or v(x, a) ∈ R, where Z or R are the integeror real number area, respectively. Without loss of generality,we assume that d1 6 d2 6 ... 6 dK . Preference relationsexist in decision table: >a, >d, 6a and 6d, which signifythe relation of no worse than or no better than with respectto a or d, respectively. For a subset of attributes B ⊆ A,we define x >B y as x >a y for all a ∈ B. We sayDT is monotone increasing consistent with respect to B, if∀x, y ∈ U , B ⊆ A, x >B y, then x >d y; otherwise, DTis monotone increasing inconsistent with respect to B. Basedon the preference relations, the x-dominated or x-dominatingsets in terms of D and B are defined. For x, y ∈ U , d ⊆ D,a ∈ A, B ⊆ A,

[x]>D = {y : y >d x}, [x]6D = {y : y 6d x}, (1)

[x]>B = {y : y >B x}, [x]6B = {y : y 6B x}. (2)

For decision di, let d>i = ∪Kj>idj and d6i = ∪Kj6idj , wheredj is the set of objects with decision dj . We have that d>i ⊇d>j and d6i ⊆ d6j for i 6 j. The lower approximations ofd>i , d

6i ⊆ U with respect to attribute subset B are defined as:

B>d>i = {y : [x]>B ⊆ d>i }, B6d6i = {y : [x]6B ⊆ d

6i }.

(3)Based on the dominance rough set theory, Qian et al. [8]

presented a discernibility matrix for monotone increasingconsistency decision table:

Definition 1: A monotone consistent discernibility matrixis defined as:

cij =

{{a ∈ A : xj ∈ [xi]

>a }, if xj ∈ [xi]

>D;

∅, otherwise.(4)

Definition 2: The discernibility function with respect to adiscernibility matrix MA(D) = {cij} is defined as:

f(MA(D)) = ∧{∨(cij)|∀i, j = 1, ..., n, cij 6= ∅}. (5)

where ∨ and ∧ are the disjunction and conjunction operator,respectively.

By transforming the disjunction norm form into the conjunc-tion norm form through the absorption law and the distributionlaw, one can obtain multiple reducts, that is, the terms of theconjunction norm form.




Considering the influence of noisy objects, Qian et al. [8]also proposed β-variable (β ∈ [0, 0.5]) upward lower andupper approximations of the decision class:

R>B(d

>i ) =

{x ∈ U |

|[x]6B ∩ d>i |

|[x]6B |> 1− β

}, (6)

R>B(d

>i ) =

{x ∈ U |

|[x]6B ∩ d>i |

|[x]6B |> β

}, (7)

where |S| is the cardinality of the set S. Based on these, thesignificance measures of the attribute a w.r.t B are defined as:

Sigβinner(a,B, d) = γβB(d)− γβB−{a}(d), (8)

Sigβouter(a,B, d) = γβB∪{a}(d)− γβB(d), (9)

where γβB(d) is defined as:

γβB(d) =

∣∣U− K⋃i=1

(R

>B(d

>i )−R>

B(d>i

)∣∣|U | .

Actually, γβB(d) measures the monotonic consistency betweenB and d, and the parameter β reflects the tolerance degree tonoise, so as to control the relaxation degree of the model.

B. Fuzzy Dominance Based Rough Sets

Let DT = (U,A ∪ {d}) be an ordinal decision table. Wedefine U/d> = {D>

i , i = 1, 2, ...,K} and U/d6 = {D6i , i =

1, 2, ...,K}, where D>i or D6

i is the membership function ofset d>i or d6i satisfying D>

i (x) ⊇ D>j (x) or D6

i (x) ⊆ D6j (x)

for i 6 j, respectively. We introduce the sigmoid functionto measure the fuzzy preference degree [32]. For x, y ∈ U ,the fuzzy dominating relation and fuzzy dominated relationbetween x and y in terms of a are defined as:

R>a (x, y) =1

1 + e−k(v(x,a)−v(y,a)), (10)

R<a (x, y) =1

1 + ek(v(x,a)−v(y,a)). (11)

It is easy to observe that R>a (x, y) +R>a (y, x) = 1. R>a (x, y)measures the preference degree of x over y: R>a (x, y) = 1

2indicates v(a, x) = v(a, y), R>a (x, y) >

12 shows v(a, x) >

v(a, y), and R>a (x, y) <12 shows v(a, x) < v(a, y). The fuzzy

dominance relation based on the sigmoid function does notsatisfy the properties of reflexivity and symmetry, but it issup-min transitive. That is:

Property 1: Let R>a be a fuzzy dominance relation basedon sigmoid function, then

(1) R>a (x, x) 6= 1;(2) R>a (x, y) 6= R>a (y, x);(3) supzmin{R>a (x, z), R>a (z, y)} 6 R>a (x, y).

The properties are also applied to R<a .The fuzzy dominance relation degrades into the dominance

relation when k → ∞. If R>a (x, y) and R<a (x, y) are notparticipated into the calculation process of learning models,the value of k dose not make any difference. The reason is thatthey are just used to compare relatively in the same dimension.

Based on R>a , the fuzzy dominating class and fuzzy domi-nated class for x in terms of a are defined as:

[x]>

a =R>a (x1, x)

x1+R>a (x2, x)

x2+ ...+

R>a (xn, x)

xn, (12)

[x]<

a =R<a (x1, x)

x1+R<a (x2, x)

x2+ ...+

R<a (xn, x)

xn. (13)

If the intersection operator is used to integrate the preferencerelations generated by multiple attributes, then for B ⊆ A, wehave:

R>B(x, y) = ∩a∈BR>a (x, y) = min

a∈BR>a (x, y), (14)

R<B(x, y) = ∩a∈BR<a (x, y) = min

a∈BR<a (x, y). (15)

Obviously, R>B(x, y) > R>A(x, y) and R<B(x, y) > R<A(x, y).The upward and downward fuzzy lower approximation in

terms of a are defined as:

R>a (D>i )(x) = inf

z∈Umax{1−R>a (z, x), D

>i (z)}, (16)

R<a (D6i )(x) = inf

z∈Umax{1−R<a (z, x), D

6i (z)}. (17)

Specifically, if D>i is the crisp set with D>

i (y) = 1 for y ∈d>i and otherwise D>

i (y) = 0. then the upward fuzzy lowerapproximation is degraded into:

R>a (D>i )(x) = inf

z 6∈d>i1−R>a (z, x). (18)

In this case, the membership of x to the lower approximationof d>i is larger if x is better than the greatest objects from theinferior classes. The larger the membership degree, the greaterthe dependence of decision attribute on feature attribute or thelarger the consistency degree between them. In this paper, weconsider the crisp ordinal decision class.

With fuzzy rough set, Hu et al. [32] considered the summa-tion of all objects’ lower approximation degrees to all decisionclasses as the approximation quality of feature attributes to thedecision attribute. Based on the idea, the upward, downwardand global significance of attribute a relative to B are definedas Eq. 19-Eq.21. Then incorporating the significance measuresinto Algorithm 1, an upward reduct, a downward reduct anda global reduct can be obtained. However, the summationoperator is not robust to noise objects, and Algorithm 1 usuallygets a reduct with some unnecessary attributes [33].

C. FREMT: Fuzzy Rank Entropy based Monotonic tree

For ordinal classification, ranking mutual information (RMI)and fuzzy ranking mutual information (FRMI) are proposedin [37]. Both RMI and FRMI are important measures ofattributes, which can be used as heuristic criterions to evaluateand to select features.

Definition 3: Given DT = (U,A ∪ D), where D = {d},for B ⊆ A ∪ D, the upward ranking mutual information ofthe set U in terms of B and D is defined as:

RMI>(B,D) = (22)

− 1

|U |

n∑i=1

log

∣∣[xi]>B∣∣× ∣∣[xi]>d ∣∣|U | ×

∣∣[xi]>B ∩ [xi]>d

∣∣ .




sig>(a,B,D) =

∑i

∑x∈d>i

(R>{B∪a}D

>i −R>BD

>i

)∑i |d

>i |

, (19)

sig<(a,B,D) =

∑i

∑x∈d6i

(R<{B∪a}D

6i −R<BD

6i

)∑i |d

6i |

, (20)

sig(a,B,D) =

∑i |d

>i |sig>(a,B,D) +

∑i |d

6i |sig<(a,B,D)∑

i |d>i +

∑i |d

6i |

. (21)

Algorithm 1 Forward greedy search for a reduct

Require: Ordinal decision table DT = (U,A ∪ {d})Ensure: An ordinal reduct of DT : B

1: B = ∅2: for each feature ai ∈ A−B, do3: Compute significance degree of attribute a by signifi-

cance measures SIG(ai, B,D) \\ such as Eq. 7, Eq.8and Eq. 19-Eq.21

4: end for5: SIG(ak, B,D) = maxi SIG(ai, B,D)6: if SIG(ak, B,D) > 0 then7: B = B ∪ ak;8: Go to Step 2;9: end if

10: END

Definition 4: Given DT = (U,A ∪ D), where D = {d},for B ⊆ A∪D, the upward fuzzy ranking mutual informationof the set U in terms of B and D is defined as:

FRMI>(B,D) = (23)

− 1

|U |

n∑i=1

log

∣∣[xi]>B∣∣× ∣∣[xi]>d ∣∣|U | ×

∣∣[xi]>B ∩ [xi]>

d

∣∣ .A monotonic decision tree based on RMI, the so-called

REMT, is established in [7]. REMT is capable to capturethe monotonic structure in ordinal classification. ReplacingRMI with FRMI in REMT, we obtain the fuzzy rank entropybased monotonic decision tree (FREMT), which is suitablefor ordinal classification and captures more information thanREMT. The algorithm of FREMT is shown as Algorithm 2.FREMT is a binary tree. The structure of FREMT is the sameas that of the famous CART algorithm, while the differenceis that CART adopts the Gini index as the heuristic criterion.

III. FUSING FUZZY MONOTONIC DECISION TREES USINGFUZZY DOMINANCE ROUGH SET

In this section, we define a discernibility matrix for ordinaldecision table using fuzzy dominance rough set and presentthe fusion algorithm for ordinal classification.

A. Discernibility Matrix with Fuzzy Dominance Rough Set

Although some significance measures of attributes havebeen proposed, the reduction methods with them are basedon the forward heuristic algorithm. Generally, the heuristic

Algorithm 2 FREMT

Require: Ordinal decision table OD = (U,A ∪ d); stopingcriterion of FREMT: ε

Ensure: A monotonic decision tree T .1: If the number of objects is 1 or all objects are from the

same class, then the branch stops growing.2: otherwise,3: for each feature ai, do4: for each cj ∈ Vai (Vai is the domain of value of ai),

do5: Divide objects into two subsets according to cj ,6: if f(ai, x) ≤ cj then7: Put x into one subset, and set f(ai, x) = 1,8: else9: Put x into the other subset, and set f(ai, x) = 2.

10: end if11: Denote the binarized feature in terms of ai

and cj as ai(cj) and compute FRMIcj =FRMI≥({ai(cj)}, {d}).

12: end for j13: c∗j = argmaxj FRMIcj .14: end for i15: Select the best feature a∗ and the corresponding point c∗:

(a∗, c∗) = argmaximaxj FRMI≥({ai(cj)}, {d}).16: If FRMI≥({a∗}, {d}) < ε, then stop.17: Build a new node and split objects with a∗ and c*.18: Recursively produce new splits according to the above

procedure until stopping criterion is satisfied.

reduction algorithm can only get a reduct and is easily affectedby the reading order of attributes. To get multiple reducts,researchers often permute the attributes [38]. This strategymakes the relation of the multiple attributes subsets unclearand lack of interpretability. In this subsection, we definea discernibility matrix for ordinal attribute reduction. Thesubsets of attributes based on the discernibility matrix arecomplete and diverse.

In the classical rough set, the union of the lower approx-imation to each decision class, the so-called positive region,is often used to define the discernibility matrix. However, inthe ordinal decision table, there is an inclusion relationshipbetween the hierarchical decision classes (d>i or d6i ), whichleads to the union of the lower approximations is equal tothe lower approximation of the universal set. Firstly, we givethe definition of the local positive region, then based on




the definition of lower approximation in [12], we obtain adiscernibility matrix.

Definition 5: For each object x, the upward local positiveregion with respect to R>A and and the downward local positiveregion with respect to R<A are defined as:

LPosR>A(D)(x) = ∪

D>i ∈U/d>,di>dx

R>A(D>i )(x), (24)

LPosR<A(D)(x) = ∪

D6i ∈U/d6,di6dx

R<A(D6i )(x). (25)

By definition, we have that LPosR>A(D)(x) takes its maxi-

mum at R>A(D>x )(x) and LPosR<

A(D)(x) takes its maximum

at R<A(D6x )(x):

LPosR>A(D)(x) = R>A(D

>x )(x), (26)

LPosR<A(D)(x) = R<A(D

6x )(x). (27)

This is consistent with the situation in the fuzzy rough setthat the positive region of an object is equal to the lowerapproximation of its own decision class [12], [14]. Comparedwith taking the summation over the lower approximationsas a significance measure, considering each object’s lowerapproximation is expected to be robust to noisy object.

Next, we will introduce another way to define the lowerapproximation, which is helpful to find the discernibilityattributes for each pair of objects. Without loss of generality,we only use the upward relation in the following.

Definition 6: The fuzzy x-dominated class in terms of R>ais defined as:

(xλ)R>a(z) =

{0, 1−R>a (z, x) > λ,

λ, 1−R>a (z, x) < λ.(28)

where λ ∈ (0, 1].In fact, (xλ)R>

ais a λ-level cut set with respect to 1−R>a .

It satisfies the following properties:1) (xλ1

)R>a⊆ (xλ2

)R>a

if λ1 < λ2;2) 1−R>A(z, x) > 1−R>B(z, x) and (xλ)R>

A⊆ (xλ)R>

Bif

B ⊆ A.Theorem 1: The lower approximation to D>

i in terms ofR>a is defined as:

R>a (D>i ) = ∪λ{(xλ)R>

a: (xλ)R>

a⊆ D>

i , λ ∈ (0, 1]}. (29)

Proof: Suppose R>a (D>i )(z) = λ∗, then we prove that

∪λ{(xλ)R>a: (xλ)R>

a⊆ D>

i , λ ∈ (0, 1]} = λ∗. By definition,we have that ∀z ∈ d<i , 1−R>a (z, x) > λ∗. Besides,

(xλ)R>a⊆ D>

i ⇔ ∀z ∈ d<i , (xλ)R>

a(z) = 0 (30)

⇔ ∀z ∈ d<i , 1−R>a (z, x) > λ. (31)

Hence, (xλ)R>a⊆ D>

i requires that λ 6 λ∗. Thus, for(xλ)R>

a⊆ D>

i , ∪λ(xλ)R>a= (xλ∗)R>

a.

In rough set and fuzzy rough set, the lower approximation ofa set can be represented by the union of multiple equivalenceclasses or fuzzy equivalence classes contained in the set anddifferent equivalence classes are mutually disjoint [12]. Indominance rough set, the lower approximation can also be ex-pressed as an union of fuzzy classes, while (xλ)R>

a6= (yλ)R>

a

⇒ (xλ)R>a∩ (yλ)R>

a= ∅ is not established. The reason is

that the transitivity of fuzzy relation is directional.

Definition 7: For a fuzzy ordinal decision table DT =(U,A∪ {d}), B ⊆ A is a local positive region reduct relativeto A if it satisfies:

1) LPosR>BD = LPosR>

AD; (32)

2) ∀a ∈ B,LPosR>{B−a}

D 6= LPosR>BD. (33)

Theorem 2: Suppose B ⊆ A is a local positive regionreduct of fuzzy ordinal decision table if and only if for eachx, B satisfies (xλ)R>

B⊆ D>

x , λ = R>A(D>x )(x).

Proof: ⇐: It is clear by using (xλ)R>A⊆ (xλ)R>

B.

Theorem 3: Suppose B ⊆ A is a local positive regionreduct of fuzzy ordinal decision table if and only if forevery x with λ = R>A(D

>x )(x), when y ∈ D<

x , we have1−R>B(y, x) > λ.

Proof: If B ⊆ A is a local positive region reduct of fuzzyordinal decision table, then for λ = R>A(D

>x )(x), we have

∀x ∈ U, (xλ)R>B⊆ D>

x , (34)

⇔ for y ∈ D<x , (xλ)R>

B(y) = 0, (35)

⇔ for y ∈ D<x , 1−R>B(y, x) > λ. (36)

According to the analysis, we can define the discernibilitymatrix MA(D) = (cij)n×n as follows:

Definition 8: A discernibility matrix keeping the local pos-itive region invariant is defined as:

cij =

{{a ∈ A : 1−R>a (xj , xi) > λi}, if xj ∈ D<

xi;

∅, otherwise,(37)

where λi = R>A(D>xi)(xi).

The above discernibility matrix makes sure of keeping thelocal positive region of each object invariant, because for∀a ∈ cij satisfies 1 − R>a (xj , xi) > λi, the final reductionB by discernibility function also satisfies 1 − R>B(xj , xi) =maxa∈B 1 − R>a (xj , xi) > λi, which meets the requirementof Theorem 3.

Next, an example is employed for showing the efficiency ofthe proposed reduction method. Table I is an ordinal decisiontable selected from the Bankruptcy risk dataset. Bankruptcyrisk records the experience of a Greek industrial developmentbank financing industrial and commercial firms. Suppose U ={x1, x2, x3, x4, x4, x5, x6} is a set of objects, each object ischaracterize by 12 ordinal feature attributes. The decision classis shown in the last row.

TABLE I: Ordinal decision table from Bankruptcy risk

U/A 1 2 3 4 5 6 7 8 9 10 11 12 Dx1 3 2 1 1 1 4 4 4 4 4 2 4 3x2 2 1 3 1 1 3 5 2 4 2 1 3 3x3 2 1 1 1 1 3 2 2 4 4 2 3 2x4 2 1 2 1 1 2 4 3 3 2 1 2 2x5 3 2 1 1 1 1 3 3 3 4 3 4 1x6 3 1 1 1 1 1 2 2 3 4 3 4 1

Based on Definition 1, the obtained discernibilitymatrix is shown in Table II. According to thediscernibility function Eq. (4), the reduction result is




TABLE II: Discernibility matrix based on Definition 1

U2 x1 x2 x3 x4 x5 x6

x1 ∅ ∅ {1, 2, 6, 7, 8, 12} {1, 2, 6, 8, 9, 10, 11, 12} {6, 7, 8, 9} {2, 6, 7, 8, 9}x2 ∅ ∅ {3, 7} {3, 6, 7, 9, 12} {3, 6, 7, 9} {3, 6, 7, 9}x3 ∅ ∅ ∅ ∅ {6, 9} {6, 9}x4 ∅ ∅ ∅ ∅ {3, 6, 7} {3, 6, 7, 8}x5 ∅ ∅ ∅ ∅ ∅ ∅x6 ∅ ∅ ∅ ∅ ∅ ∅

TABLE III: Discernibility matrix based on Definition 8

U2 x1 x2 x3 x4 x5 x6

x1 ∅ ∅ {7, 8} {6, 10, 12} {6} {6, 7, 8}x2 ∅ ∅ {3, 7} {3, 6, 7, 9, 12} {3, 6, 7, 9} {3, 6, 7, 9}x3 ∅ ∅ ∅ ∅ {6} {6}x4 ∅ ∅ ∅ ∅ {3, 6, 7} {3, 6, 7, 8}x5 ∅ ∅ ∅ ∅ ∅ ∅x6 ∅ ∅ ∅ ∅ ∅ ∅

{{7, 9}, {6, 7}, {3, 9, 12}, {3, 8, 9}, {3, 6}, {2, 3, 9}, {1, 3, 9}}.The less monotone consistency attributes {1} and {12} areincluded in the result by sample pair (x1, x3), (x1, x4)and (x2, x4). While in the proposed discernibility matrix(Definition 8), each attribute is judged by λi, which is thevalue with respect to the most monotone consistent attribute:

λi = R>A(D>xi)(xi) = inf

z∈d<xi

maxa∈A

1−R>a (z, xi). (38)

Thus, the attributes will make a comparison with each otherbefore selection. Our discernibility matrix is shown in TableIII and the reduction result is {{6, 7}, {3, 7, 8}}, shown asEq. (39). This signifies that the proposed discernibility matrixmakes full use of the global information of attributes.

Besides, the reduction result of Algorithm 1 with Eq. (19) asthe significance measure is {6, 1, 7}. Obviously, the attribute{1} should not be included in. However, after {6} is selectedinto the reduction set, {1} increases the membership degreeof x5 and x6 to d>3 , which leads to the increase of the totaldegree. This signifies that the summation operator is not robustto the noise objects, which may produce an inappropriatereduct.

B. A More General Discernibility Matrix for Ordinal Classi-fication

Because the preference relationship induced by an attributeset is determined by the intersection of the relationshipsinduced by its elements, as shown as Eq. (15) and Eq. (16), thelocal lower approximation for each object λi is the distance interms of the most discriminative attribute with the best objectsfrom its inferior class. Thus in Definition 8, the attributeswhich are not inferior to the best attribute can be selectedinto the discernibility matrix. This requirement may be strictfor real-world classification problems. For classification, evenif an attribute produces a small gap for objects, it mayprovide some discrimination information. To contain the weakmonotonicity of attributes and the inconsistency of decisiontable, we use the quantile operator 1 − p instead of the

intersection operator to integrate the multiple attributes. Thus,the maximization operator in λi is generalized to the p-thmaximization.

Definition 9: A generalized discernibility matrix for ordinalclassification is defined as:

cij =

{{a ∈ A : 1−R>a (xj , xi) > λi}, if xj ∈ D<

xi;

∅, otherwise,(40)

where λi = infz∈d<xi[1− R>a (z, xi), a ∈ A]p and [·]p denotes

the p-th maximal value in set.In practice, we use Definition 9 to build the discernibility

matrix. The parameter p controls the number of attributesselected into cij , and thus influences the number of attributesin the reducts. When p value is 1, Definition 9 is degraded toDefinition 8. This is the most rigorous way to filter attributes.In this case, the number of attributes in cij is less, and theprobability that the intersection of cij is empty is larger, thusthe probability that the reducts have more attributes is larger.When the p value approaches the number of original features,all the attributes will be selected for each pair of objects, andthe reducts are singleton sets.

C. Algorithms for Fusing Complete Fuzzy Monotonic Deci-sion Trees

After building the discernibility matrix, we will find reductsbased on it. Skowron and Rauszer proposed a reductionmethod with a discernibility function [34], which can findall the reducts simultaneously. The essential of this methodis the laws of absorption and distribution. Here, we presentefficient algorithms to implement the laws of absorption anddistribution. Next, following a discernibility matrix simplifica-tion methodology for constructing attribute reducts proposed in[39], we offer an efficient procedure to find a reduct based onthe discernibility matrix. Finally, we give our fusion algorithm.

Based on the absorption law in discernibility function, onlythe minimum elements M∗A(D) that cannot be contained byother elements in the discernibility matrix are sufficient andnecessary to find the final reduction [40]. For example, in TableII, only {7∨ 8}, {6} and {3∨ 7} are required. Putting all theelements into the discernibility function definitely increasesthe time complexity and the computational load.

To compress discernibility matrix, Chen et.al. proposedthe SPS method [41], which selects the sample pairs thatdetermine the minimum elements. The SPS is a point-basedalgorithm and its time complexity is |U |4|A|. Here, we presenta vector-based algorithm to implement the absorption law.Firstly, we represent the inverse of the discernibility matrix




f(MA(D)) ={7 ∨ 8} ∧ {6 ∨ 10 ∨ 12} ∧ {6} ∧ {6 ∨ 7 ∨ 8} ∧ {3 ∨ 7} ∧ {3 ∨ 6 ∨ 7 ∨ 9 ∨ 12} (39)∧ {3 ∨ 6 ∨ 7 ∨ 9} ∧ {3 ∨ 6 ∨ 7} ∧ {3 ∨ 6 ∨ 7 ∨ 8}

={7 ∨ 8} ∧ {6} ∧ {3 ∨ 7} = {6 ∧ 7} ∨ {3 ∧ 6 ∧ 8}.

as a binary matrix. Here, the inverse refers to the De Morganlaw. For example, the inverse of {6 ∨ 7} ∧ {3 ∨ 6 ∨ 8} is{6 ∧ 7} ∨ {3 ∧ 6 ∧ 8}. Let the rows and the columns in thebinary matrix correspond to the disjunction operator ∨ andthe conjunction operator ∧, respectively. Thus the inverse ofMA(D) is represented as:

MA(D)ij =

{1, the ith term contains the jth attribute;0, otherwise.

MA(D) is a matrix with |A| columns and at most |U |2rows because there are empty sets in the discernibility matrix.Algorithm 3 is designed to execute the absorption law bydeleting the other rows that contain the current one. The timecomplexity of Algorithm 3 is O(L∗L|A|), where L∗ and L arethe numbers of rows in M

∗A(D) and MA(D), respectively and

L∗ ≤ L ≤ |U |2. The worst case time complexity of Algorithm3 is the same as the SPS method, while Algorithm 3 runs fasterthan SPS in the real application because there are not too manycounting, intersection and union operations in Algorithm 3.

Algorithm 3 Matrix absorption law

Require: The inverse of the original discernibility matrixMA(D)

Ensure: The inverse of the absorbed discernibility matrixM∗A(D)

1: Find the unique rows of MA(D);2: Sort the rows of MA(D) in ascending order in terms of

the number of 1′s in each row and represent the sortedmatrix as M

∗A(D);

3: Let L∗ be the number of rows in M∗A(D) and i = 1;

4: while i ≤ L∗ do5: Delete the other rows that contain the ith row

in M∗A(D), that is, the other rows that satisfyM∗A(D)CTi = li, where Ci is the ith row in M

∗A(D)

and li is the number of 1′s in Ci;6: Update L∗

7: i = i+ 18: end while

Next, we implement the distribution law in the discernibil-ity function based on Shannon’s expansion [42]. Shannon’sexpansion shows that any matrix or disjunction normal formM can be represented by a combination of two sub-functionsof the original:

M = mjMmj +Mmj ,

where mj is the variable in j-th column, Mmjis obtained by

setting the j-th column as 0 in M , Mmjis obtained by deleting

the rows that the j-th column is 1 in M , and Mmj is theinverse of Mmj

. By recursively using Shannon’s expansion,

Algorithm 4 Matrix distribution law: Shannon’s expansion forM∗A(D)

Require: The inverse of the absorbed discernibility matrixM∗A(D)

Ensure: A set of reducts RED1: Choose the rows in M

∗A(D) which have the fewest

number of 1’s and put the features that appear in theserows into feature subset A0;

2: Choose the attribute from A0 that appears most often inthe other rows in M

∗A(D) (suppose it is the jth attribute);

3: Build the matrix of the right branch Rightj with the rowsin M

∗A(D) whose jth column are 0;

4: Build the matrix of the left branch Leftj by setting thejth column of M

∗A(D) as 0 and finding the unique of the

rows;5: if There is a row of all 0′s in Leftj then6: Left′j = ∅;7: else8: Left′j ← Algorithm 4 (Leftj);9: end if

10: if Rightj = ∅ then11: Right′j is equal to the zero vector with |A| columns;12: else if Rightj has only one row then13: Let l be the number of 1′s in Rightj ;14: Initialize Right′j as the zero matrix with l rows and |A|

columns;15: for i = 1 to l do16: Denote h as the column index corresponding to the

ith 1 of Rightj ;17: Set the ith row and hth column in Right′j as 1;18: end for19: else20: Right′j ← Algorithm 4 (Rightj);21: end if22: Set the jth column in Right′j as 1;23: RED = [Left′j ;Right

′j ].

the conjunction normal MA(D) can be transformed to thedisjunction normal form, and the reducts can be found. Thedetailed implementation processes are shown in Algorithm 4.Actually, the right branch of and the left branch of Algorithm4 implement the mjMmj

and Mmj, respectively. One can

refer to Ref. [42] for a vivid example. The time complexityof Algorithm 4 is O(L∗V1), where V1 is the number of nodesand L∗ is the time complexity of each node (the number ofrows in M

∗A(D)).

Algorithm 4 finds all reducts simultaneously. Generally, fordata sets with a high dimension, the number of reducts foundby Algorithm 4 is quite large. However, in ensemble learning,




fewer base classifiers can achieve satisfactory performance.Thus, sometimes we need to manually specify the number ofbase classifiers for fusion. Based on these two considerations,it is very meaningful to find a reduct by discernibility ma-trix. In [39], Yao and Zhao proposed a discernibility matrixsimplification methodology for constructing attribute reducts.Following their work, we give a procedure to find a reductbased on discernibility matrix, which is shown as Algorithm 5.This algorithm executes the absorption and deletion operationsrow by row to simplify the elements in discernibility matrixto singleton sets, and the union of all the singleton sets isa reduct. The deletion operation complies with the fact thatthe attributes set A can be deleted if none of the subsets ofA corresponds to a row in M

∗A(D). The time complexity of

Algorithm 5 is O(tL∗|A|), where t is the number of iterationsand t� L∗. Running Algorithm 5 multiple times can generatea set of reducts for learning multiple monotonic decision trees.

Algorithm 5 Finding a reduct based on discernibility matrix

Require: The inverse of the absorbed discernibility matrixM∗A(D)

Ensure: A reduct B1: Sort the rows of M

∗A(D) randomly;

2: Let L∗ be the number of rows in M∗A(D);

3: while i ≤ L∗ do4: Select the ith row form M

∗A(D) and note as mi;

5: Put the features that appear in mi into features set A;6: Randomly select a feature from A, note as a;7: Set the ath column of mi to be 1 and other columns to

be 0;8: Set A0 = A− a9: for j = i+ 1 to L∗ do

10: Let mj be the jth row in M∗A(D);

11: if The ath column of mj is 1 then12: Delete mj ;13: else if The features that appear in mj are not a subset

of A0 then14: Set the A0 column of mj as 0;15: end if16: end for17: Update L∗ and set i = i+ 1;18: end while19: Output B as the union of attribute that appears in M

∗A(D).

Based on the above two methods to find reducts, we developtwo fusion methods. One is for using all reducts (Algorithm4), and the other is for fusion with a specified number ofbase classifiers (Algorithm 5). Using each reduct, we grow aFREMT and fuse all FREMT trees by majority voting. Thatis, an object is classified into the class that is assigned bythe majority of the FREMT trees. For the fusion strategy, wecan also use the median of the label as the prediction label,while the vote strategy is a more robust and lower-complexityapproach [28]. The detailed algorithm is shown in Algorithm6, named as FFREMT: Fusing Fuzzy Rank Entropy BasedMonotonic Tree. The FFREMT mainly contains two aspects:generating attribute reduction subsets and fusing multiple

REMT trees. As for the first aspect, the time complexity hasbeen given above. The time complexity of building L′ FREMTtrees is O(L′bV2|U ||A|2), where b is the number of nodes inthe tree and V2 is the number of different values in the features.

Algorithm 6 FFREMT

Require: Ordinal decision table OD = (U,A ∪ d); stopingcriterion of FREMT: ε; Parameter of discernibility matrixp; Sample to be predicted: x

Ensure: The decision of x1: Generate discernibility matrix MA(D) by Definition 9

with parameter p and express it in binary matrix form;2: Simplify MA(D) as M

∗A(D) by Algorithm 3;

3: Find multiple reducts RED with M∗A(D) by Algorithm

4 or by running Algorithm 5 L′ times; \\ Algorithm 4is designed for finding all reducts and Algorithm 5 for areduct.

4: Transform each row of RED into the collection form anddenote as {B1, ..., BL′}, where L′ is the rows number ofRED;

5: for B1 to BL′ do6: learn a monotonic decision tree Tl with Algorithm 2.7: make a prediction on x using Tl: dl(x) = Tl(x).8: end for9: Return the predict class by voting rule: d(x) =

argmaxk

( L′∑j=1

I{dl = k})

(I{·} is the counting function).

IV. EXPERIMENTAL ANALYSIS

In this section, the effectiveness of FFREMT is shown bycomparing with several other ensemble methods for ordinalclassification.

A. Benchmark Methods and Data Sets

FREMT [7] is used as the base classifier for all ensemblemethods. Four significance measures for ordinal attributes havebeen defined in [8] and [32], as introduced in Eq. (8)-(9),(19), (20) and (21) in Section II. The algorithm for searchingmultiple reduction subsets with significance measures havebeen applied in [8], that is, Algorithm 3 in [8]. We embed thefour significance measures into the Algorithm 3 in [8] to obtainmultiple reductions and use them construct multiple FREMT,and the ensemble methods based on the four significancemeasures are represented as β-red [8], up-sig [32], down-sig[32], and global-sig [32], respectively. In addition, there aretwo discernibility matrixes defined for ordinal classification.One is the monotonic consistency matrix MCM in Definition1. The other is FCMT based on the monotonic consistency set[22]. We embed MCM and FCMT into the Step 1 of Algorithm6 to fuse FREMT. We also use bagging to promote FREMT. Inbagging, we form 21 new training sets, construct FREMT onthem and ensemble the trees by voting rules. We will compareFFREMT with these methods.




TABLE IV: Data sets in the experimental analysis

ID Data sets objects features classes source1 Breast cancer 699 9 2 UCI2 Wine-red 1599 12 6 KEEL3 Wine 178 13 3 KEEL4 Boston housing 506 13 4 WEKA5 Australian 690 14 2 UCI6 German 1000 19 2 KEEL7 Student score 512 25 3 [8]8 Statlog Landsat Satellite 6435 36 7 UCI9 Waveform Database Generator40 5000 40 3 UCI10 Molecular Biology 106 57 2 UCI11 Residential-Building 372 103 5 UCI12 Urban Land Cover 168 147 9 UCI13 Parkinson’s Disease 756 752 2 UCI14 DrivFace 606 6400 3 UCI15 Arcene 100 10000 2 UCI

Fifteen real-world classification tasks are used as benchmarkdata sets. The detail information of them is described inTable IV. The missing values in data sets are imputed bythe nearest-neighbor method, while the rows or columns withtoo many missing values will be deleted. The predict variableof Residential-Building is continuous. We discretize it by thebin method with a ratio of 144:105:90:19:14. For the firstseven data sets, we find all the reducts by Algorithm 4, andcompare FFREMT with all of the above mentioned benchmarkmethods. For the last eight data sets, we find 21 reductsby Algorithm 5, and because the significance measure basedmethods are computationally infeasible, we only compareFFREMT with Bagging and the other two discernibility matrixbased methods.

B. Data Pre-processingTo ensure that the preference degree generated by each

attribute is in the same dimension, a data set is normalizedthrough training set by the range method. That is, eachattribute is normalized by the following transformation:

v(x, a) =v(x, a)−miny∈U1

v(y, a)

maxy∈U1 v(y, a)−miny∈U1 v(y, a), (41)

where U1 is the training set.In practice, the data set may present decreasing monotonic-

ity: the worse feature value gets the better decision class.In this case, we need to preprocess each attribute in datasets to satisfy the assumption of increasing monotonicity.There are several solutions to this problem. Here, we choosethe Spearman rank correlation coefficient to measure themonotonicity consistency between attribute and decision. Inparticular, let ai and di (i = 1, 2, · · · , n) be the ranking resultson n objects according to attribute and decision, respectively.The rank correlation coefficient is calculated as:

Spearman(a, d) = 1− 6

n(n2 − 1)

n∑i=1

(ai − di)2. (42)

If the Spearman rank correlation coefficient between it withthe decision is negative, we use one minus the attribute toreplace the original one.

Besides, label noise is also widespread and makes impactson the decision-making. In ordinal decision table with increas-ing monotonic constraint, the objects with a better attribute buta worse decision class will be considered as a noisy object.Although we have preprocessed the data to satisfy increasingmonotonic, this process is totally based on a global Spearmanrank correlation coefficient value. Here, we reset the label oftraining objects which belong to a monotone inconsistent set.In particular, let

UβM ={x| |[x]

≤A∩[x]

≤d |

|[x]≤A |≥ 1− β}, if |[x]≥A| < n0;

{x| |[x]≥A∩[x]

≥d |

|[x]≥A |≥ 1− β}, if |[x]≤A| < n0;

{x| |[x]≥A∩[x]

≥d |

|[x]≥A |≥ 1− β, |[x]

≤A∩[x]

≤d |

|[x]≤A |≥ 1− β}, otherwise

(43)

be the monotone consistent set, where β ∈ [0, 0.5] and n0 is aconstant that is much smaller than the number of the trainingobjects. The probability that an object belongs to dk can beestimated by:

P (d(x) = dk) = P (d(x) ≥ dk)− P (d(x) ≥ dk+1), (44)

where

P(d(x) > dk

)=

∣∣[x]>A ∩ d>k ∣∣∣∣[x]>A∣∣ , (45)

and P(d(x) > dK+1

)= 0. For x ∈ U1 − UβM , the label of it

should be reset as:

d(x)∗ = argmaxk

P (d(x) = dk), k = 1, ...,K. (46)

C. Evaluation Measures and Parameter Selection

Here, the classification accuracy CA and the mean absoluteerror MAE are employed to evaluate the performance of themethods. CA is computed as:

CA =1

n

n∑i=1

I{d(xi) = d(xi)}, (47)

where I{·} is the counting function: if d(xi) = d(xi), thenI{d(xi) = d(xi)} = 1, otherwise I{d(xi) = d(xi)} = 0.MAE is computed as:

MAE =1

n

n∑i=1

|d(xi)− d(xi)|, (48)

where n is the number of objects, d(xi) is the predict decisionand d(xi) is the true decision.

For the first seven data sets, we randomly divide it into U1

and a test set U2 at a ratio of 7:3, and for the last eight biggerones, we divide at a ratio of 1:9. We conduct the division24 times to obtain the average performance. All methods arerun under the same U1-U2 division. For the methods withoutparameters to tune, we use U1 as the training set. For themethods with parameters to tune, we further divide U1 intoa training set U11 and a validation set U12 at a ratio of




FREMT up down global MCM β -red FCMT Bagging OUR-20

-10

0

10

20

30

40

Th

e g

ap

of

sig

nif

ica

nt

win

an

d s

ign

ific

an

t lo

ss

MAE

FREMT up down global MCM β -red FCMT Bagging OUR-15

-10

-5

0

5

10

15

20

25

30

Th

e g

ap

of

sig

nif

ica

nt

win

an

d s

ign

ific

an

t lo

ss

CA

Fig. 1: Significance Test Results in terms of MAE and CA

7:3. We conduct the U11-U12 division 10 times to obtainthe mean validation MAE for each pair of parameters. Andthe parameters with the minimum mean validation MAE arechosen as the optimal parameter for training. The parameterp is chosen from 1 to min{b|A|/2c, 10} with a step 1 and βis chosen from 0.1 to 0.5 with a step 0.1.

D. Experimental Results and Analysis

Table V and VI records the mean and standard deviationof test MAE and test CA. In each row, the method with themaximum value is in bold font; black dots mark behind thevalues of a method which FFREMT is significantly better thanunder the pairwise right-tailed Student’s test with 95 percentsignificance level. The last row records the times of win-lose-tie by comparing FFREMT with the others, and the numbersin brackets record the times of significant win. It can beobserved that FFREMT performs better on most data sets, andmakes significant improvements in Wine-red, Boston housing,German, Student Score, Stalog, Waveform and Arcene.

To further analyze the results, we compare these methodsby their confidence intervals [23]. Let [L(a,d,m), U(a,d,m)] bethe 95 percent confidence interval of method a on data set din the sense of performance measure m:

L(a,d,m) = µ(a,d,m) − 1.96σ(a,d,m)√

n,

U(a,d,m) = µ(a,d,m) + 1.96σ(a,d,m)√

n,

(49)

where µ(a,d,m) is the mean value and σ(a,d,m) is the standarddeviation across n times runs. For two methods a and a′,if La,d,m > Ua′,d,m, then we say that a significantly winsa′, otherwise a significantly loses to a′. Each bar in Fig. 1represents the number gap between the significant win andlose times of a given method compared with the others onthe first seven data sets. From Fig. 1, we can see that theperformance of FFREMT is significantly better than the othersix reduction-based ensemble methods and Bagging.

We show the influence of parameters on each data set inFig. 2. The x-axis and the y-axis represent the value of β andp, respectively. The z-axis represents mean validation MAE.We can observe that the parameter p plays an important rolein tuning the performances of the FFREMT algorithm. ForBreast cancer, Wine, Student score and Stalog, a larger p valueperforms better. This signifies that for data sets with moremonotonic consistency attributes, the parameter p should belarger. For Boston housing, Australian, German and Arcene,the influence of parameter β is more than that of parameter p.

We also show the comparison of mean reduction time andmean tree-building time on each data set in Fig. 3. It is easy

to observe that the reduction methods based on significancemeasures are more time consuming than the discernibilitymatrix based methods. And in discernibility matrix basedmethods, the time is dominated by tree-building, which ismeet the basic requirements of ensemble learning. Thesedemonstrate that the proposed reduction method and ensemblemethod is effective and serviceable.

V. CONCLUSION AND FUTURE WORK

In this paper, we mainly focus on developing a fusionmethod based on attribute reduction to solve ordinal classifica-tion problem. To achieve this, we firstly define a discernibilitymatrix with fuzzy dominance rough set and introduce thealgorithms for finding reducts based on it. Each reduct formsa feature subspace with original information, and ordinaldecision trees are built in these feature subspaces. Finally,we fuse these trees by voting. Numerical experimental resultsdemonstrate that the proposed fusion method is effectiveand feasible. The major contribution of this paper is thatwe have theoretically defined a discernibility matrix usingfuzzy dominance rough set and have proposed an effectivefusion method for ordinal classification problem. We haveverified that the reducts based on discernibility matrix are wellsuited to ensemble learning, and the combination of them iscompetent to further improve the generalization performanceof the fuzzy monotonic decision tree. In addition, we haveoffered efficient procedures for implementing the laws ofabsorption and distribution, and for finding a reduct basedon the discernibility matrix. In the future, we will workon eliminating the random consistency from the attributedependence measure to describe the attribute more objectivelyand correctly. We will also consider attribute reduction andensemble methods for mixture decision tables which containboth ordinal and nominal attributes.

ACKNOWLEDGMENT

This work was supported by National Key R&D Programof China (No. 2018YFB1004300), National Natural ScienceFoundation of China(Nos. 61672332, 61872226, 61802238,61976120), Program for the Outstanding Innovative Teams ofHigher Learning Institutions of Shanxi, Program for the SanJin Young Scholars of Shanxi, the Overseas Returnee ResearchProgram of Shanxi Province (No. 2017023), Natural ScienceFoundation of Shanxi Province (Grant No. 201701D121052).

REFERENCES

[1] L. A. Zadeh, “Fuzzy sets,” Information & Control, vol. 8, no. 3, pp.338–353, 1965.

[2] D. Dubois and H. Prade, “Rough fuzzy sets and fuzzy rough sets,”International Journal of General Systems, vol. 17, no. 2-3, pp. 191–209, 1990.

[3] Z. Pawlak, “Rough sets,” International Journal of Computer and Infor-mation Sciences, vol. 11, no. 5, pp. 341–356, 1982.

[4] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzzyrough sets,” Fuzzy Sets and Systems, vol. 126, no. 2, pp. 137–155, 2002.

[5] D. S. Yeung, D. Chen, E. C. C. Tsang, J. W. T. Lee, and W. Xizhao,“On the generalization of fuzzy rough sets,” IEEE Transactions on FuzzySystems, vol. 13, no. 3, pp. 343–361, 2005.

[6] R. Jensen and Q. Shen, “Fuzzy rough attribute reduction with applicationto web categorization,” Fuzzy Sets and Systems, vol. 141, no. 3, pp. 469–485, 2004.




TAB

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V:

Com

pari

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AE

and

CA

DA

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MA

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std

[7]

[32]

[8]

[22]

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10.

0581±

0.01

38•

0.05

79±

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37•

0.05

67±

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73•

0.05

94±

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16•

0.04

74±

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06•

0.07

54±

0.10

130.

0442±

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730.

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370.

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96

20.

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97•

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56±

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0.34

12±

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0.20

42±

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68•

0.19

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63•

0.16

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07

30.

0796±

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85•

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47•

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990.

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270.

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0.39

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670.

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920.

3883±

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540.

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310.

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74

50.

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0.19

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0.01

880.

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390.

1869±

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140.

1931±

0.02

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0.18

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1633±

0.02

970.

1777±

0.02

71

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3855±

0.02

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93±

0.01

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89±

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20•

0.38

08±

0.03

03•

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16±

0.02

38•

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51±

0.03

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31

70.

2557±

0.03

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0.03

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88•

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81±

0.02

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0.02

200.

1634±

0.02

39•

0.10

97±

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87

win

-los

e-tie

7(7)

-0-0

7(6)

-0-0

7(6)

-0-0

7(5)

-0-0

6(4)

-1-0

7(5)

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6(2)

-0-1

6(4)

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DA

TAID

CA±

std

[7]

[32]

[8]

[22]

[27]

OU

R

FRE

MT

up-s

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sig

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al-s

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β-r

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Bag

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10.

9419±

0.01

38•

0.94

21±

0.01

37•

0.94

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0.01

72•

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0.01

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0.92

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130.

9558±

0.01

730.

9565±

0.01

370.

9589±

0.00

97

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7875±

0.01

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0.78

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0.78

44±

0.01

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64±

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0.01

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0.01

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0.01

07

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9250±

0.04

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0.03

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0.03

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0.03

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980.

9174±

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9204±

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87

win

-los

e-tie

7(7)

-0-0

7(7)

-0-0

7(6)

-0-0

7(6)

-0-0

6(4)

-1-0

7(5)

-0-0

7(2)

-0-0

6(4)

-1-0




0.0410

0.045

0.6

0.05

Valid

ati

on

MA

E

0.055

p

5 0.4

β

0.06

0.20 0

(a) Breast cancer

0.166

0.165

0.6

Valid

ati

on

MA

E

4

0.17

p

0.4

β

0.175

2 0.20 0

(b) Wine-red

0.0415

0.06

0.6

0.08

Valid

ati

on

MA

E

10

0.1

p

0.4

β

0.12

5 0.20 0

(c) Wine

0.310

0.4

0.6

0.5

Valid

ati

on

MA

E

0.6

p

5 0.4

β

0.7

0.20 0

(d) Boston housing

0.17510

0.18

0.6

0.185

Valid

ati

on

MA

E

0.19

p

5 0.4

β

0.195

0.20 0

(e) Australian

0.3310

0.34

0.6

0.35

Valid

ati

on

MA

E

0.36

p

5 0.4

β

0.37

0.20 0

(f) German

0.110

0.15

0.6

0.2

Valid

ati

on

MA

E

0.25

p

5 0.4

β

0.3

0.20 0

(g) Student score

0.3510

0.4

0.45

0.6

Valid

ati

on

MA

E

0.5

p

0.55

5 0.4

β

0.6

0.20 0

(h) Statlog

0.310

0.35

0.6

0.4

Valid

ati

on

MA

E

0.45

p

5 0.4

β

0.5

0.20 0

(i) Waveform

0.210

0.25

0.6

Valid

ati

on

MA

E

0.3

p

5 0.4

β

0.35

0.20 0

(j) Molecular Biology

0.1910

0.2

0.6

0.21

Valid

ati

on

MA

E

0.22

p

5 0.4

β

0.23

0.20 0

(k) Residential-Building

0.810

1

0.6

Valid

ati

on

MA

E

1.2

p

5 0.4

β

1.4

0.20 0

(l) Urban Land Cover

0.210

0.21

0.22

0.6

Valid

ati

on

MA

E

0.23

p

0.24

5 0.4

β

0.25

0.20 0

(m) Parkinson’s Disease

0.0810

0.1

0.6

0.12

Valid

ati

on

MA

E

0.14

p

5 0.4

β

0.16

0.20 0

(n) DrivFace

0.3210

0.34

0.6

0.36

Valid

ati

on

MA

E

0.38

p

5 0.4

β

0.4

0.20 0

(o) Arcene

Fig. 2: The Influence of Parameters on Validation MAE

FREMT up down global MCM β -red FCMTBagging OUR0

10

20

30

40

50

60

70

Tim

e(s

)

Tree-building Time

Reduction Time

(a) Breast cancerFREMT up down global MCM β -red FCMTBagging OUR

0

20

40

60

80

100

120

140

160

180

200

Tim

e(s

)

Tree-building Time

Reduction Time

(b) Wine-redFREMT up down global MCM β -red FCMTBagging OUR

0

10

20

30

40

50

60

Tim

e(s

)

Tree-building Time

Reduction Time

(c) WineFREMT up down global MCM β -red FCMTBagging OUR

0

20

40

60

80

100

120

Tim

e(s

)

Tree-building Time

Reduction Time

(d) Boston housingFREMT up down global MCM β -red FCMTBagging OUR

0

50

100

150

200

250

Tim

e(s

)Tree-building Time

Reduction Time

(e) Australian

FREMT up down global MCM β -red FCMTBagging OUR0

200

400

600

800

1000

1200

Tim

e(s

)

Tree-building Time

Reduction Time

(f) GermanFREMT up down global MCM β -red FCMTBagging OUR

0

100

200

300

400

500

600

700

800

900

Tim

e(s

)

Tree-building Time

Reduction Time

(g) Student scoreFREMT MCM FCMT Bagging OUR

0

200

400

600

800

1000

Tim

e(s

)

Tree-building Time

Reduction Time

(h) StatlogFREMT MCM FCMT Bagging OUR

0

200

400

600

800

1000

1200

Tim

e(s

)

Tree-building Time

Reduction Time

(i) WaveformFREMT MCM FCMT Bagging OUR

0

10

20

30

40

50

Tim

e(s

)

Tree-building Time

Reduction Time

(j) Molecular Biology

FREMT MCM FCMT Bagging OUR0

200

400

600

800

1000

1200

1400

Tim

e(s

)

Tree-building Time

Reduction Time

(k) Residential-BuildingFREMT MCM FCMT Bagging OUR

0

100

200

300

400

500

Tim

e(s

)

Tree-building Time

Reduction Time

(l) Urban Land CoverFREMT MCM FCMT Bagging OUR

0

20

40

60

80

100

120

140

Tim

e(s

)

Tree-building Time

Reduction Time

(m) Parkinson’s DiseaseFREMT MCM FCMTBagging OUR

0

2000

4000

6000

8000

10000

12000

Tim

e(s

)

Tree-building Time

Reduction Time

(n) DrivFaceFREMT MCM FCMT Bagging OUR

0

1000

2000

3000

4000

5000

6000

7000

Tim

e(s

)

Tree-building Time

Reduction Time

(o) Arcene

Fig. 3: Time Comparison




TABLE VI: Comparison on MAE and CA

DATA IDMAE ± std

FREMT MCM FCMT Bagging FFREMT

8 0.5933 ±0.0396 • 0.3718 ±0.0218 0.3703 ±0.0188 0.3712 ±0.0132 0.3644 ±0.0180

9 0.4341 ±0.0165 • 0.4632 ±0.0293 • 0.4585 ±0.0258 • 0.3901 ±0.0195 • 0.3377 ±0.0360

10 0.4625 ±0.1100 • 0.3563 ±0.0646 • 0.3438 ±0.0442 • 0.3625 ±0.0754 • 0.2406 ±0.0590

11 0.2426 ±0.0207 • 0.2139 ±0.0479 0.2078 ±0.0415 0.2035 ±0.0253 0.1852 ±0.0396

12 1.5128 ±0.2321 • 0.8964 ±0.2757 0.9508 ±0.2691 1.0709 ±0.2385 • 0.8236 ±0.2384

13 0.2722 ±0.0237 • 0.2394 ±0.0068 • 0.2299 ±0.0100 • 0.2106 ±0.0126 0.2082 ±0.0125

14 0.1924 ±0.0479 • 0.1063 ±0.0118 0.1030 ±0.0081 0.1062 ±0.0161 0.1030 ±0.0133

15 0.4194 ±0.0974 • 0.3419 ±0.0531 0.3452 ±0.0628 0.3226 ±0.0938 0.3000 ±0.0900

win-lose-tie 8(8)-0-0 8(3)-0-0 8(3)-0-0 8(3)-0-0

DATA IDCA ± std

FREMT MCM FCMT Bagging FFREMT

8 0.7547 ±0.0143 • 0.8514 ±0.0069 0.8516 ±0.0069 0.8523 ±0.0047 0.8526 ±0.0072

9 0.6656 ±0.0127 • 0.6633 ±0.0212 • 0.6686 ±0.0157 • 0.7079 ±0.0159 • 0.7448 ±0.0257

10 0.5375 ±0.1100 • 0.6438 ±0.0646 • 0.6562 ±0.0442 • 0.6375 ±0.0754 • 0.7594 ±0.0590

11 0.7617 ±0.0214 • 0.7991 ±0.0411 0.8052 ±0.0392 0.8000 ±0.0249 0.8174 ±0.0408

12 0.5255 ±0.0714 • 0.7509 ±0.0752 0.7273 ±0.0647 0.6546 ±0.0781 • 0.7564 ±0.0708

13 0.7278 ±0.0237 • 0.7607 ±0.0068 • 0.7701 ±0.0100 • 0.7894 ±0.0126 0.7918 ±0.0125

14 0.8153 ±0.0454 • 0.8964 ±0.0093 0.8984 ±0.0068 0.8944 ±0.0162 0.8993 ±0.0111

15 0.5807 ±0.0974 • 0.6581 ±0.0531 0.6548 ±0.0628 0.6774 ±0.0938 0.7000 ±0.0900

win-lose-tie 8(8)-0-0 8(3)-0-0 8(3)-0-0 8(3)-0-0

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Jieting wang received a B.S. and M.S. degree atschool of Mathematical Sciences from Shanxi Uni-versity, China, in 2013 and 2015, respectively. She isa PhD candidate at Institute of Big Data Science andIndustry, Shanxi University. Her research interestincludes statistical machine learning and ensemblelearning.

Yuhua Qian received the M.S. and Ph.D. degrees incomputers with applications from Shanxi University,Taiyuan, China, in 2005 and 2011, respectively.

He is currently a Professor with the Key Labora-tory of Computational Intelligence and Chinese In-formation Processing, Ministry of Education, ShanxiUniversity. He is best known for multigranulationrough sets in learning from categorical data andgranular computing. He is involved in research onpattern recognition, feature selection, rough set the-ory, granular computing, and artificial intelligence.

He has authored over 80 articles on these topics in international journals.He served on the Editorial Board of the International Journal of Knowledge-Based Organizations and Artificial Intelligence Research. He has served as theProgram Chair or Special Issue Chair of the Conference on Rough Sets andKnowledge Technology, the Joint Rough Set Symposium, and the Conferenceon Industrial Instrumentation and Control, and a PC Member of many machinelearning, data mining, and granular computing conferences.

Feijiang Li received a B.S. degree at school ofComputer Science and Technology from NortheastUniversity, China, in 2012. He is a PhD candidate atInstitute of Big Data Science and Industry, ShanxiUniversity. His research interest includes machinelearning and knowledge discovery.

Jiye Liang received the M.S. and Ph.D. degreesfrom Xi’an Jiaotong University, Xi’an, China, in1990 and 2001, respectively. He is currently a Pro-fessor with the School of Computer and Informa-tion Technology, Shanxi University, Taiyuan, China,where he is also the Director of the Key Laboratoryof Computational Intelligence and Chinese Informa-tion Processing of the Ministry of Education. Hehas authored more than 170 journal papers in hisresearch fields. His current research interests includecomputational intelligence, granular computing, data

mining, and knowledge discovery.

Weiping Ding (M’16-SM’19) received the Ph.D.degree in Computation Application, Nanjing Uni-versity of Aeronautics and Astronautics (NUAA),Nanjing, China, in 2013. He was a Visiting Scholarat University of Lethbridge (UL), Alberta, Canada,in 2011. From 2014 to 2015, He is a PostdoctoralResearcher at the Brain Research Center, NationalChiao Tung University (NCTU), Hsinchu, Taiwan.In 2016, He was a Visiting Scholar at National Uni-versity of Singapore (NUS), Singapore. From 2017to 2018, he was a Visiting Professor at University

of Technology Sydney (UTS), Ultimo, NSW, Australia. His main researchdirections involve granular computing data mining, machine learning and bigdata analytics. He has published over 50 papers in flagship journals andconference proceedings as the first author, including IEEE Transactions onFuzzy Systems, IEEE Transactions on Neural Network and Learning System,IEEE Transactions on Cybernetics, and so on. To data, he has held 12 approvedinvention patents in total over 20 issued patents. Dr. Ding currently serveson the Editorial Advisory Board of Knowledge-based Systems. He serves asan Associate Editor of IEEE Transactions on Fuzzy Systems, InformationSciences, Swarm and Evolutionary Computation, Journal of Intelligent &Fuzzy Systems, and IEEE Access, and a leading guest editor for fourinternational journals. He serves/served as a program committee member forseven international conferences and workshops. Dr. Ding is a member of TaskForce on Adaptive and Evolving Fuzzy Systems of IEEE CIS FSTC, a memberof Soft Computing TC of IEEE SMC, a member of Granular Computing (GrC)TC of IEEE SMC, and also a member of Data Mining and Big Data AnalyticsTC of IEEE CIS.

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