CN103646118A - Confidence dominance-based rough set analysis model and attribute reduction methods - Google Patents

Abstract

The invention discloses a confidence dominance-based rough set analysis model and two attribute reduction methods, which are applicable to solving the incomplete preference decision problem and discovers the more important attribute to a decision under consistent or inconsistent information. The invention provides a new expanded dominance relation, namely confidence dominance relation which obeys order relation characteristics including reflexivity, transitivity and order symmetry, compared with the existing expanded dominance relation, the confidence dominance relation can avoid semantic conflict, and the approximation precision and classification accuracy of the approximate confidence dominance-based rough set analysis model are more excellent through theorem proving and instance analysis. In addition, in order to find out the more important attributes of the decision and aiming at the incompletely consistency and inconsistency conditions, the invention also discloses the two attribute reduction methods under the confidence dominance relation, including an identification matrix-based attribute reduction method and a classification precision-based heuristic attribute reduction method.

Description

Put letter dominance based rough set model and attribute reduction method

Technical field

The present invention relates to data attribute reduction method, particularly a kind of letter dominance based rough set model and attribute reduction method put.

Background technology

Along with the fast development of computer networking technology, the data volume of every field increases rapidly, but because the reasons such as restriction, transmission fault and some human factors of data acquisition technology cause data defect and Loss often to occur.In real world, due to complicacy and the uncertainty of environment, people by the face of information is uncertain, not exclusively and the decision problem with preferential information.

Rough set theory is based upon on relation of equivalence (meeting reflexivity, symmetry and transitivity) basis, is a kind of new mathematical tool of processing uncertain and ambiguous information.Dominance based rough set (Dominance-based Rough Set Analysis, DRSA) utilizes dominance relationship (meeting reflexivity and transitivity) to substitute relation of equivalence, and classical rough set is expanded to the orderly infosystem with decision-making preference.For incomplete, information has deletion condition, and multiple expansion relation is suggested the decision problem that solution information is uncertain, incomplete and have preferential information.

Extended dominance relation, meets reflexivity and transitivity, but definition is too loose, allows two objects not have common non-null attribute to compare.Limited Extended Dominance Relation, limits two objects and has at least a common non-null attribute to compare, the situation of avoiding two objects not have common non-null attribute to compare, and limited extended model meets reflexivity.Generalized extended dominance relation, by threshold value, limit the number of two comparable common non-null attributes of object, avoiding two objects to have too many attribute is the situation that missing values still can compare, Generalized extended dominance relation is the special case of Limited Extended Dominance Relation, but generalized extended model does not meet reflexivity.Degree extended attribute dominance relationship is refined as missing values " loss " and " being indifferent to " two kinds of situations under Generalized extended dominance relation, but still does not meet reflexivity.Similar dominance relationship missing values is considered as uncertain, non-existent value, cannot be described, and only allows relatively given value part, and missing values part does not allow comparison.Similar dominance relationship meets reflexivity and transitivity, but dominance relationship and inferior position relation adopt two kinds of form of Definitions, make but to differ and obtain surely.Restriction dominance relationship, the dominance relationship of missing values object is limited in the situation of attribute maximal value and minimum value, limit similar dominance relationship in conjunction with restriction dominance relationship and similarity relation, these two definition all need to know maximal value and the minimum value of property value, and definition is too strict, missing values object can only could be determined restriction dominance relationship in attribute maximal value or minimum value situation.

Above-mentioned various expansion dominance relationships are proposed for processes incomplete order information, but the characteristic of " order " system that is all wanting in consideration exists the semantic contradiction under some sequence characteristics.

The present invention proposes order relation characteristic, meets reflexivity, transitivity and order symmetry.Therefore,, in incomplete Order Strategic Decision problem, need to propose a kind of new expansion dominance relationship and meet order relation characteristic.

Summary of the invention

In view of this, technical matters to be solved by this invention is to provide a kind of letter dominance based rough set model and attribute reduction method put.

The object of the present invention is achieved like this:

Letter dominance based rough set model and the attribute reduction method put provided by the invention, comprises the following steps:

S1: obtaining information data are also set up decision system DS according to the information data of obtaining;

S2: judge in decision system, whether all properties value exists missing values, if so, set up incomplete Order Strategic Decision system IODS;

S3: build and put the definition of letter dominance relationship according to incomplete Order Strategic Decision system IODS;

S4: build rough set model according to putting the definition of letter dominance relationship;

S5: judge that according to rough set model whether incomplete Order Strategic Decision system IODS is consistent decision system, if so, adopts the attribute reduction method based on identification matrix;

S6: if not, adopt the heuristic attribute reduction method based on nicety of grading.

Further, the definition of described decision system DS and the definition of putting letter dominance relationship meet following relational expression:

Definition 1: be provided with a decision system DS=(U, A, V, f);

Wherein,

U is domain, i.e. the object set of non-NULL;

A is community set, A=C ∪ D, and wherein, C and D represent respectively conditional attribute set decision kind set;

V is attribute codomain, has preference;

F:U * A → V is information function, f={f (x _i, a) | f (x _i, a): x _i→ v _a, a ∈ C, x _i∈ U, 1≤i≤| U|}, wherein, x _iobject in expression field, a represents certain conditional attribute, v _athe value that represents attribute a, the mapping of → representative function, f (x _i, a)=v _aindicated object x _ivalue on attribute a.

If all property values are all known, be called complete Order Strategic Decision system; If there is missing values, be called incomplete Order Strategic Decision system IODS;

Definition 2: suppose x, y ∈ U,

b _p(x)={ b|b ∈ P ∧ f (x, b) ≠ * }, puts letter dominance relationship (Confidential Dominance Relation, CDR) and is defined as follows:

CDR(P)={(x,y)∈U ²||B _P(x)∩B _P(y)|/|B _P(x)|=1)∧

Wherein, x, y represents the object in domain, P represents the subset of conditional attribute set C, B _p(x) indicated object x property value is not empty community set, and CDR (P) represents to put the set of letter dominance relationship under property set P, uses

represent " y puts letter advantage in x ";

Definition 3: supposition DS=(U, A, V, f) is an IODS, x ∈ U, the letter advantage of the putting collection of x is: $D_p^{\mathrm{CDR}+}\left(x\right)=\{y\∈U|y{D}_P^{\mathrm{CDR}}x\};$ The letter inferior position collection of putting of x is defined as follows: $D_p^{\mathrm{CDR}-}\left(x\right)=\{y\∈U|x{D}_P^{\mathrm{CDR}}y\};$

Character 1

(1) put letter dominance relationship and meet reflexivity, transitivity and order symmetry;

（2） $G=\left\{D_p^{\mathrm{CDR}+}\left(x\right)\right|x\∈U\}$ And $G=\left\{D_p^{\mathrm{CDR}-}\left(x\right)\right|x\∈U\}$ All coverings of U;

(3) if $x_j\∈D_p^{\mathrm{CDR}+}\left(x_i\right),D_p^{\mathrm{CDR}+}\left(x_j\right)\⊆D_p^{\mathrm{CDR}+}\left(x_i\right);$

If $x_j\∈D_p^{\mathrm{CDR}-}\left(x_i\right),D_p^{\mathrm{CDR}-}\left(x_j\right)\⊆D_p^{\mathrm{CDR}-}\left(x_i\right).$

Further, the definition of described rough set model meets following relational expression:

Suppose that DS=(U, A, V, f) is an IODS, decision attribute D is divided into n class Cl={C by domain U _lt|t ∈ 1,2..., n}}, wherein

in t class, combine and be defined as

under combine and be defined as

show that x at least belongs to class Cl _t,

show that x belongs to class Cl at the most _t; Upper and lower approximate, Boundary Region, approximation quality and nicety of grading based on putting letter dominance relationship are defined as follows:

Definition 4:

x ∈ U, t=1,2 ..., n,

with

upper and lower be approximately defined as follows respectively:

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\{x\∈U|D_p^{\mathrm{CDR}+}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≥}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≥}}{\∪}{D}_p^{\mathrm{CDR}+}\left(x\right)=\{x|D_p^{\mathrm{CDR}+}\left(x\right)\∩{\mathrm{Cl}}_t^{\≥}\≠\mathrm{\Φ}\};$

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\{x\∈U|D_p^{\mathrm{CDR}-}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≤}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≤}}{\∪}{D}_p^{\mathrm{CDR}-}\left(x\right)=\{x|D_p^{\mathrm{CDR}-}\left(x\right)\∩{\mathrm{Cl}}_t^{\≤}\≠\mathrm{\Φ}\};$

Definition 5: Boundary Region is defined as ${\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≥}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right),{\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≤}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right);$

Definition 6:

with

coarse approximate approximation quality be defined as follows respectively:

$\mathrm{\α}\left({\mathrm{Cl}}_t^{\≥}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|,\mathrm{\α}\left({\mathrm{Cl}}_t^{\≤}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|;$

Definition: 7: nicety of grading is defined as follows:

${\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≥}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≥}\right)|/\left|U\right|,{\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≤}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≤}\right)|/\left|U\right|;$

Wherein, Cl represents the set of the classification that decision attribute is divided domain, Cl _trepresent t Decision Classes, t represents classification subscript, in like manner, and Cl _nrepresent n Decision Classes, Cl _n-1represent n-1 Decision Classes, Cl _srepresent s Decision Classes,

represent not to be inferior to the object intersection of t Decision Classes,

represent not to be better than the object intersection of t Decision Classes, associating in expression

lower approximate set,

the letter advantage of the putting class of indicated object x, associating in expression

boundary Region, associating in expression

upper approximate set,

represent lower associating

lower approximate collection,

represent lower associating

upper approximate collection,

associating in expression

approximation quality,

represent lower associating

approximation quality, associating in expression

nicety of grading,

represent lower associating nicety of grading, Φ represents empty set.

Further, describedly judge that whether incomplete Order Strategic Decision system IODS is consistent decision system and carries out giving a definition by meeting:

Definition 8: supposition DS=(U, A, V, f) is an IODS, if

claim that decision system DS is consistent; Otherwise, claim that decision system DS is incomparable inconsistent;

Definition 9: supposition decision system DS=(U, A, V, f) is consistent, if

and

to arbitrarily

have

claim that P is a yojan of decision system DS;

Definition 10: supposition x, y ∈ U, D ^*={ (x, y): f (D, x)>=f (D, y) }, the resolution property set of x and y is defined as follows:

the identification matrix that is called decision system DS;

Definition 11: identification function definition is

this function has determined the yojan of incomplete consistent decision system DS;

Wherein, CDR (C) represents to put letter dominance relationship collection under conditional attribute C, and CDR (D) represents to put letter dominance relationship collection under decision attribute D, the value of f (D, x) indicated object x on decision attribute D, f (D, y) represent the value of indicated object y on decision attribute D

indicated object x, the distinctive function of y,

represent identification matrix, △ ^*represent yojan result, ∧ represents conjunction computing, and ∨ represents the computing of extracting.

Further, the described attribute reduction method based on identification matrix comprises the following steps:

S51: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S52: the identification matrix of obtaining decision system DS according to definition 10

S53: according to definition 11 yojan of obtaining about decision system DS.

Further, the described heuristic attribute reduction method based on nicety of grading comprises the following steps:

S61: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S62: obtain the upper and lower approximate collection of decision system DS, obtain the Boundary Region of decision system DS, calculating nicety of grading is γ _c;

S63: make i=1;

S64:C'=C-{a _i, a wherein _i∈ C;

S65: recalculate C' nicety of grading γ _c';

S66: if γ _c≤ γ _c', remove attribute a _i, C=C', otherwise C'=C;

S67: if i=|C| finishes; Otherwise, i=i+1, execution step 4;

Wherein, U represents field, and x represents domain object, and i represents cycle count, and C represents conditional attribute combination, and C' represents to remove the conditional attribute collection after certain attribute, | C| represents the base of conditional attribute collection, a _irepresent i conditional attribute, γ _cbeing illustrated in conditional attribute integrates as the nicety of grading under C, γ _c'be illustrated in the nicety of grading under the set of C' conditional attribute.

The invention has the advantages that: the present invention has adopted and put letter dominance based rough set model and 2 kinds of attribute reduction methods, is applicable to solve incomplete Preference Decision problem, under consistent information or inconsistent information, find the prior attribute of decision-making.The present invention proposes a kind of new expansion dominance relationship, put letter dominance relationship, defer to order relation characteristic, be reflexivity, transitivity and order symmetry, compare with existing expansion dominance relationship, can avoid contradiction semantically, by theorem proving and instance analysis, approximation quality and the nicety of grading of putting letter dominance based rough set approximate model that the present invention proposes are more excellent.In addition, for finding out the prior attribute of decision-making, for the incomplete one inconsistent situation of making peace, two kinds of attribute reduction methods under letter dominance relationship are put in invention, respectively the attribute reduction method based on identification matrix, and the heuristic attribute reduction method based on nicety of grading.

Accompanying drawing explanation

In order to make the object, technical solutions and advantages of the present invention clearer, below in conjunction with accompanying drawing, the present invention is described in further detail, wherein:

Fig. 1 puts letter dominance based rough set model and attribute reduction method process flow diagram for what the embodiment of the present invention provided.

Embodiment

Below with reference to accompanying drawing, the preferred embodiments of the present invention are described in detail; Should be appreciated that preferred embodiment is only for the present invention is described, rather than in order to limit the scope of the invention.

Embodiment 1

The present embodiment is first introduced the existing dominance relationship of expanding:

Definition 12: extended dominance relation,

with

represent.Restriction dominance relationship is the special shape of definition 12, only allows to expand in attribute maximal value and minimum value.

Definition 13: Limited Extended Dominance Relation,

With

represent.

Definition 14: Generalized extended dominance relation

With

represent.K degree extended attribute dominance relationship is the special shape of definition 14, missing values is subdivided into two kinds of situations and expands.

Definition 15: similar dominance relationship

$\begin{array}{c}{\mathrm{SDR}}^{+}\left(P\right)=\{(x,y)\∈U^2:\∀q\∈B_P,f(x,q)\≥f(y,q)\}\\ {\mathrm{SDR}}^{-}\left(P\right)=\{(x,y)\∈U^2:\∀q\∈B_P,f(x,q)\≤f(y,q)\}\end{array}$

With

with

represent.Restriction similarity relation is by the special shape of restriction relation and definition 15 combinations.

The contrast of several expansion dominance relationships

Order relation Character Comparison

Under order infosystem, order relation characteristic should have reflexivity, transitivity and order symmetry.Several expansion dominance relationships meet reflexivity, transitivity and ordered pair and claim implementations as shown in Table 1.

The character contrast of several relations of table one

We illustrate several expansion relations not meeting for the semantic contradiction under order infosystem and order relation characteristic by way of example.Suppose to have five object x ₁=[1, *, 2, *], x ₂=[2, *, *, 1], x ₃=[2, *, 1, *], x ₄=[2, *, 2,1], x ₅=[*, 2, *, 2].

(1) from definition, can obtain

and

_?x ₅>=x ₁and x ₅≤ x ₁, from the characteristic of order, be not difficult to obtain x ₅=x ₁, but be difficult to obtain this conclusion from data object.

(2)

but

and be false, therefore

do not meet transitivity.If x ₃>=x ₂and x ₂>=x ₁, in order relation, can obtain x ₃>=x ₁, and cannot can not get x in Limited Extended Dominance Relation ₃>=x ₁conclusion.

(3) although

be

special shape, threshold value λ get zero situation, but

but likely cannot meet reflexivity, if establish λ >0.5, the x existing in order relation ₁>=x ₁, in Generalized extended dominance relation, but not necessarily meet,

be false.

(4) from similar dominance relationship definition, can obtain, but

do not meet, this presents semantic contradiction under order infosystem, meets x ₄>=x ₁, but can not obtain x ₁≤ x ₄.

Put letter dominance relationship and meet reflexivity, transitivity and three order relation characteristics of order symmetry, " y puts letter advantage in x ",

on the non-null attribute of x, y is all better than x, and the quantity of information of y is no less than x, and definition has met the characteristic of order and met real dominance relationship semantic, has avoided the semantic contradiction that several expansion relations produce above.

Embodiment 2

Fig. 1 puts letter dominance based rough set model and attribute reduction method process flow diagram for what the embodiment of the present invention provided, as shown in the figure: letter dominance based rough set model and the attribute reduction method put provided by the invention, comprises the following steps:

S1: obtaining information data are also set up decision system DS according to the information data of obtaining;

S2: judge in decision system, whether all properties value exists missing values, if so, set up incomplete Order Strategic Decision system IODS;

S3: build and put the definition of letter dominance relationship according to incomplete Order Strategic Decision system IODS;

S4: build rough set model according to putting the definition of letter dominance relationship;

S5: judge that according to rough set model whether incomplete Order Strategic Decision system IODS is consistent decision system, if so, adopts the attribute reduction method based on identification matrix;

S6: if not, adopt the heuristic attribute reduction method based on nicety of grading.

The definition of described decision system DS and the definition of putting letter dominance relationship meet following relational expression:

Definition 1: be provided with a decision system DS=(U, A, V, f);

Wherein,

U is domain, i.e. the object set of non-NULL;

A is community set, A=C ∪ D, and wherein, C and D represent respectively conditional attribute set decision kind set;

V is attribute codomain, has preference;

F:U * A → V is information function, f={f (x _i, a) | f (x _i, a): x _i→ v _a, a ∈ C, x _i∈ U, 1≤i≤| U|}, wherein, x _iobject in expression field, a represents certain conditional attribute, v _athe value that represents attribute a, the mapping of → representative function, f (x _i, a)=v _aindicated object x _ivalue on attribute a.

If all property values are all known, be called complete Order Strategic Decision system; If there is missing values, be called incomplete Order Strategic Decision system IODS;

Definition 2: suppose x, y ∈ U,

b _p(x)={ b|b ∈ P ∧ f (x, b) ≠ * }, puts letter dominance relationship (Confidential Dominance Relation, CDR) and is defined as follows:

CDR(P)={(x,y)∈U ²||B _P(x)∩B _P(y)|/|B _P(x)|=1)∧

Wherein, x, y represents the object in domain, P represents the subset of conditional attribute set C, B _p(x) indicated object x property value is not empty community set, and CDR (P) represents to put the set of letter dominance relationship under property set P, uses

represent " y puts letter advantage in x ";

Definition 3: supposition DS=(U, A, V, f) is an IODS, x ∈ U, the letter advantage of the putting collection of x is: $D_p^{\mathrm{CDR}+}\left(x\right)=\{y\∈U|y{D}_P^{\mathrm{CDR}}x\};$ The letter inferior position collection of putting of x is defined as follows: $D_p^{\mathrm{CDR}-}\left(x\right)=\{y\∈U|x{D}_P^{\mathrm{CDR}}y\};$

Character 1

(1) put letter dominance relationship and meet reflexivity, transitivity and order symmetry;

（2） $G=\left\{D_p^{\mathrm{CDR}+}\left(x\right)\right|x\∈U\}$ And $G=\left\{D_p^{\mathrm{CDR}-}\left(x\right)\right|x\∈U\}$ All coverings of U;

(3) if $x_j\∈D_p^{\mathrm{CDR}+}\left(x_i\right),D_p^{\mathrm{CDR}+}\left(x_j\right)\⊆D_p^{\mathrm{CDR}+}\left(x_i\right);$

If $x_j\∈D_p^{\mathrm{CDR}-}\left(x_i\right),D_p^{\mathrm{CDR}-}\left(x_j\right)\⊆D_p^{\mathrm{CDR}-}\left(x_i\right).$

The definition of described rough set model meets following relational expression:

Suppose that DS=(U, A, V, f) is an IODS, decision attribute D is divided into n class Cl={Cl by domain U _t| t ∈ { 1,2..., n}}, wherein Cl _n> Cl _n-1> ... > Cl ₁, in t class, combine and be defined as under combine and be defined as

show that x at least belongs to class Cl _t,

show that x belongs to class Cl at the most _t; Upper and lower approximate, Boundary Region, approximation quality and nicety of grading based on putting letter dominance relationship are defined as follows:

Definition 4:

x ∈ U,

t=1,2 ..., n,

with

upper and lower be approximately defined as follows respectively:

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\{x\∈U|D_p^{\mathrm{CDR}+}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≥}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≥}}{\∪}{D}_p^{\mathrm{CDR}+}\left(x\right)=\{x|D_p^{\mathrm{CDR}+}\left(x\right)\∩{\mathrm{Cl}}_t^{\≥}\≠\mathrm{\Φ}\};$

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\{x\∈U|D_p^{\mathrm{CDR}-}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≤}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≤}}{\∪}{D}_p^{\mathrm{CDR}-}\left(x\right)=\{x|D_p^{\mathrm{CDR}-}\left(x\right)\∩{\mathrm{Cl}}_t^{\≤}\≠\mathrm{\Φ}\};$

Definition 5: Boundary Region is defined as ${\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≥}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right),{\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≤}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right);$

Definition 6:

with

coarse approximate approximation quality be defined as follows respectively:

$\mathrm{\α}\left({\mathrm{Cl}}_t^{\≥}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|,\mathrm{\α}\left({\mathrm{Cl}}_t^{\≤}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|;$

Definition: 7: nicety of grading is defined as follows:

${\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≥}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≥}\right)|/\left|U\right|,{\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≤}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≤}\right)|/\left|U\right|;$

Wherein, Cl represents the set of the classification that decision attribute is divided domain, Cl _trepresent t Decision Classes, t represents classification subscript, in like manner, and Cl _nrepresent n Decision Classes, Cl _n-1represent n-1 Decision Classes, Cl _srepresent s Decision Classes, represent not to be inferior to the object intersection of t Decision Classes,

represent not to be better than the object intersection of t Decision Classes, associating in expression

lower approximate set,

the letter advantage of the putting class of indicated object x,

associating in expression

boundary Region,

associating in expression upper approximate set,

represent lower associating

lower approximate collection,

represent lower associating

upper approximate collection,

associating in expression

approximation quality,

represent lower associating

approximation quality, γ _p(Cl ^>=) associating in expression

nicety of grading, γ _p(Cl ^≤) the lower associating of expression

nicety of grading, Φ represents empty set.

Describedly judge that whether incomplete Order Strategic Decision system IODS is consistent decision system and carries out giving a definition by meeting:

Definition 8: supposition DS=(U, A, V, f) is an IODS, if

claim that decision system DS is consistent; Otherwise, claim that decision system DS is incomparable inconsistent;

Definition 9: supposition decision system DS=(U, A, V, f) is consistent, if

and

to arbitrarily have

claim that P is a yojan of decision system DS;

Definition 10: supposition x, y ∈ U, D ^*={ (x, y): f (D, x)>=f (D, y) }, the resolution property set of x and y is defined as follows:

the identification matrix that is called decision system DS;

Definition 11: identification function definition is this function has determined the yojan of incomplete consistent decision system DS;

Wherein, CDR (C) represents to put letter dominance relationship collection under conditional attribute C, and CDR (D) represents to put letter dominance relationship collection under decision attribute D, the value of f (D, x) indicated object x on decision attribute D, f (D, y) represent the value of indicated object y on decision attribute D indicated object x, the distinctive function of y,

represent identification matrix, △ ^*represent yojan result, ∧ represents conjunction computing, and ∨ represents the computing of extracting.

The described attribute reduction method based on identification matrix comprises the following steps:

S51: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S52: the identification matrix of obtaining decision system DS according to definition 10

S53: according to definition 11 yojan of obtaining about decision system DS.

The described heuristic attribute reduction method based on nicety of grading comprises the following steps:

S61: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S62: obtain the upper and lower approximate collection of decision system DS, obtain the Boundary Region of decision system DS, calculating nicety of grading is γ _c;

S63: make i=1;

S64:C'=C-{a _i, a wherein _i∈ C;

S65: recalculate C' nicety of grading γ _c';

S66: if γ _c≤ γ _c', remove attribute a _i, C=C', otherwise C'=C;

S67: if i=|C| finishes; Otherwise, i=i+1, execution step 4;

Wherein, U represents field, and x represents domain object, and i represents cycle count, and C represents conditional attribute combination, and C' represents to remove the conditional attribute collection after certain attribute, | C| represents the base of conditional attribute collection, a _irepresent i conditional attribute, γ _cbeing illustrated in conditional attribute integrates as the nicety of grading under C, γ _c'be illustrated in the nicety of grading under the set of C' conditional attribute.

Embodiment 3

The difference of the present embodiment and embodiment 2 is only:

Embodiment provided by the invention provides puts letter dominance based rough set model and attribute reduction method, comprises the following steps:

S1: the definition of putting letter dominance relationship;

S2: the rough set model based on putting letter dominance relationship;

S3: in incomplete consistent information table, the attribute reduction method based on identification matrix.

S4: in incomplete inconsistent information table, the heuristic attribute reduction method based on nicety of grading.

The definition of putting letter dominance relationship (Confidential Dominate Relation, CDR), meets order relation characteristic, meets reflexivity, transitivity and order symmetry; Putting the advantage class object that letter dominance relationship defines certain object A should be more excellent than the known attribute of A, and Given information amount can not be lower than object A.

Based on putting letter dominance based rough set model, compare with the existing dominance based rough set model of expanding, approximation quality and nicety of grading are more excellent.For incomplete consistent information table, the definition of identification matrix and attribute reduction method.For incomplete inconsistent information, keep nicety of grading not reduce, heuristic attribute reduction method.

Several coarse approximate contrasts based on expanding dominance relationship

Theorem 1:

（1） $\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}$

（2） $\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}\⊆\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\⊆\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}$

Proof: (1) because $\∀x,y\∈U,y{D}_P^{\mathrm{CDR}}x\⇒y{D}_P^{\mathrm{GEDom}}x\⇒y{D}_P^{\mathrm{EDom}}x,$ Otherwise be false, therefore can obtain

from lower approximate definition, $\∀x\∈\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}\⇒x\∈\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\⇒x\∈\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}},$ Otherwise be false, so $\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}$ Set up, card is finished.

(2) proving by the same methods.

Theorem 2

（1） $\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{EDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{LEDom}}\⊆\underset{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{CDR}}$

（2） $\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{CDR}}\⊆\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{LEDom}}\⊆\stackrel{\‾}{P}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{EDom}}$

Proof: method of proof is the same, provable.

Inference 1

（1） ${\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}\⊆{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\⊆{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}$

（2） ${\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{CDR}}\⊆{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{LEDom}}\⊆{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{EDom}}$

Proof: can directly obtain by theorem 2 and theorem 3.

Above theorem and deduction, from theoretical proof put letter dominance relationship and the existing relation of expanding between dominance relationship.

Further the approximate classification performance between them is contrasted below.

Theorem 4

（1） $\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}\≥\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\≥\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}$

（2） $\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{CDR}}\≥\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{LEDom}}\≥\mathrm{\α}{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{EDom}}$

Proof: by theorem 2, theorem 3 can directly obtain.

Theorem 4 explanations, put letter dominance relationship and on coarse approximation quality, are better than the coarse approximation quality under extended dominance relation, Limited Extended Dominance Relation and Generalized extended dominance relation.

Theorem 5

（1） ${\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{CDR}}\≥{\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{GEDom}}\≥{\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≥}\right)}^{\mathrm{EDom}}$

（2） ${\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{CDR}}\≥{\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{LEDom}}\≥{\mathrm{\γ}}_P{\left({\mathrm{Cl}}_t^{\≤}\right)}^{\mathrm{EDom}}$

Proof: can directly obtain by inference 1.

Theorem 5 explanations, put letter dominance relationship and in nicety of grading, are better than the classification approximation precision under extended dominance relation, Limited Extended Dominance Relation and Generalized extended dominance relation.

The information table of certain school Development of teaching management information system of take is below put the concrete methods of realizing of letter dominance based rough set model and attribute reduction method as example explanation.

Table two is an information table of certain school Development of teaching management information system, is obviously an IODS.Wherein, course is respectively by a ₁, a ₂and a ₃represent, in table 1,2,3 submeters represent excellent, in, poor.Attribute d represents the general comment to student.

Certain school teaching general comment table of table two

Student ID	a 1	a 2	a 3	d
					1	2	1	1	1
2	3	2	*	3
					3	2	*	*	1
4	*	2	2	3
					5	*	3	1	1
6	3	2	1	3

From putting letter dominance relationship and coarse approximate definition easily obtains:

$D_P^{\mathrm{CDR}+}\left(1\right)=\left\{\mathrm{1,6}\right\};D_P^{\mathrm{CDR}+}\left(2\right)=\left\{\mathrm{2,6}\right\};D_P^{\mathrm{CDR}+}\left(3\right)=\left\{\mathrm{1,2,3,6}\right\};D_P^{\mathrm{CDR}+}\left(4\right)=\left\{4\right\};D_P^{\mathrm{CDR}+}\left(5\right)=\left\{5\right\};$ $D_P^{\mathrm{CDR}+}\left(6\right)=\left\{6\right\}.$

$D_P^{\mathrm{CDR}-}\left(1\right)=\{1,3\};D_P^{\mathrm{CDR}-}\left(2\right)=\left\{\mathrm{2,3}\right\};D_P^{\mathrm{CDR}-}\left(3\right)=\left\{3\right\};D_P^{\mathrm{CDR}-}\left(4\right)=\left\{4\right\};D_P^{\mathrm{CDR}-}\left(5\right)=\left\{5\right\};$ $D_P^{\mathrm{CDR}-}\left(6\right)=\left\{\mathrm{1,2,3,6}\right\}.$

${\mathrm{Cl}}_3^{\≥}=\left\{\mathrm{2,4,6}\right\};{\mathrm{Cl}}_1^{\≥}=\left\{\mathrm{1,2,3,4,5,6}\right\};{\mathrm{Cl}}_1^{\≤}=\left\{\mathrm{1,3,5}\right\};{\mathrm{Cl}}_3^{\≤}=\left\{\mathrm{1,2,3,4,5,6}\right\}.$

$\underset{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{CDR}}=\left\{\mathrm{2,4,6}\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{CDR}}=\left\{\mathrm{2,4,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{CDR}}=\mathrm{\Φ};\mathrm{\α}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{CDR}}=1;{\mathrm{\γ}}_P{\left({\mathrm{Cl}}^{\≥}\right)}^{\mathrm{CDR}}=1$

$\underset{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{CDR}}=\left\{\mathrm{1,3,5}\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{CDR}}=\left\{\mathrm{1,3,5}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{CDR}}=\mathrm{\Φ};\mathrm{\α}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{CDR}}=1;{\mathrm{\γ}}_P{\left({\mathrm{Cl}}^{\≤}\right)}^{\mathrm{CDR}}=1.$

Upper and lower approximate collection, Boundary Region, approximation quality and the nicety of grading of dominance relationship, Limited Extended Dominance Relation and Generalized extended dominance relation equally can be expanded.The threshold value λ value 2/3 of Generalized extended dominance relation, because its threshold value is higher, approximation quality and nicety of grading are higher.

$\underset{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{EDom}}=\left\{1\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{EDom}}=\left\{\mathrm{1,2,3,4,5,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{EDom}}=\left\{\mathrm{2,3,4,5,6}\right\};$ $\mathrm{\α}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{EDom}}=0.17;$ γ _P(Cl ^≤) ^EDom=0.17。

$\underset{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{EDom}}=\mathrm{\Φ};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{EDom}}=\{2,,\mathrm{3,4,5,6}\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{EDom}}=\left\{\mathrm{2,3,4,5,6}\right\};\mathrm{\α}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{EDom}}=0;$ γ _P(Cl ^≥) ^EDom=0.17

$\underset{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{LEDom}}=\left\{\mathrm{1,3}\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{LEDom}}=\left\{\mathrm{1,2,3,4,5,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{LEDom}}=\left\{\mathrm{2,5,6}\right\};\mathrm{\α}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{LEDom}}=0.4;$ γ _P(Cl ^≤) ^LEDom=0.5。

$\underset{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{LEDom}}=\left\{4\right\};\stackrel{\‾}{p}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{LEDom}}=\left\{\mathrm{2,4,5,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{LEDom}}=\left\{\mathrm{2,5,6}\right\};\mathrm{\α}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{LEDom}}=0.25;$ γ _P(Cl ^≥) ^LEDom=0.5

$\underset{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{1,3}\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{1,3,5,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{5,6}\right\};\mathrm{\α}{\left({\mathrm{Cl}}_1^{\≤}\right)}^{\mathrm{GEDom}}=0.5;$ γ _P(Cl ^≤) ^LEDom=0.667。

$\underset{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{2,4}\right\};\stackrel{\‾}{P}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{2,4,5,6}\right\};{\mathrm{Bn}}_P{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{GEDom}}=\left\{\mathrm{5,6}\right\};\mathrm{\α}{\left({\mathrm{Cl}}_3^{\≥}\right)}^{\mathrm{GEDom}}=0.5;$ γ _P(Cl ^≥) ^GEDom=0.667

Obviously meet theorem and deduction previously discussed, also further illustrate and put that letter dominance relationship is coarse approximate to be compared with existing method, can obtain larger positive domain space, less Boundary Region, for this reason, coarse approximate have higher approximation quality and the nicety of grading based on putting letter dominance relationship.

Attribute reduction

(1) the identification matrix attribute reduction method based on putting letter dominance relationship

For the method is better described, select the table two in example to describe as an example.The letter dominance relationship collection of putting of this example provides as front, and the identification matrix of structure is tried to achieve according to definition, as shown in Table 3:

Table three identification matrix

?

1

2

3

4

5

6

1

Φ

Φ

Φ

Φ

a 2

Φ

2

a 3

Φ

Φ

a 3

a 2a 3

a 3

3

a 2a 3

Φ

Φ

Φ

a 2a 3

Φ

4

a 1

a 3

a 2a 3

Φ

a 2

Φ

5

a 1a 2

Φ

a 2a 3

Φ

Φ

Φ

6

Φ

Φ

Φ

a 3

a 2

a 1

According to definition 11, to identification matrix is every, calculate, calculate and try to achieve yojan RED={a ₁, a ₂, a ₃.

(2) the heuristic attribute reduction method based on nicety of grading

For explanation the method, we still adopt the table two in example to describe as an example.In example each object put letter dominance relationship collection, upper and lower approximate and Boundary Region is all as front given.

Therefore, at attribute C={a ₁, a ₂, a ₃, time γ _c(Cl ^>=) ^cDR=1.

Remove the attribute a in C ₁after, portion calculates and show that nicety of grading is

do not meet heuristic evidence, retain a ₁;

Remove the attribute a in C ₂, the nicety of grading calculating

satisfy condition, therefore remove a ₂, C=C-{a ₂;

Remove the attribute a in C ₃, the nicety of grading calculating do not satisfy condition, reserved property a ₃.

Finally, all properties calculates complete, the attribute reduction result RED={a obtaining ₁, a ₃.

The foregoing is only the preferred embodiments of the present invention, be not limited to the present invention, obviously, those skilled in the art can carry out various changes and modification and not depart from the spirit and scope of the present invention the present invention.Like this, if within of the present invention these are revised and modification belongs to the scope of the claims in the present invention and equivalent technologies thereof, the present invention is also intended to comprise these changes and modification interior.

Claims (6)

1. put letter dominance based rough set model and attribute reduction method, it is characterized in that: comprise the following steps:

S1: obtaining information data are also set up decision system DS according to the information data of obtaining;

S2: judge in decision system, whether all properties value exists missing values, if so, set up incomplete Order Strategic Decision system IODS;

S3: build and put the definition of letter dominance relationship according to incomplete Order Strategic Decision system IODS;

S4: build rough set model according to putting the definition of letter dominance relationship;

S5: judge that according to rough set model whether incomplete Order Strategic Decision system IODS is consistent decision system, if so, adopts the attribute reduction method based on identification matrix;

S6: if not, adopt the heuristic attribute reduction method based on nicety of grading.

2. letter dominance based rough set model and the attribute reduction method put according to claim 1, is characterized in that: the definition of described decision system DS and the definition of putting letter dominance relationship meet following relational expression:

Definition 1: be provided with a decision system DS=(U, A, V, f);

Wherein,

U is domain, i.e. the object set of non-NULL;

A is community set, A=C ∪ D, and wherein, C and D represent respectively conditional attribute set decision kind set;

V is attribute codomain, has preference;

F:U * A → V is information function, f={f (x _i, a) | f (x _i, a): x _i→ v _a, a ∈ C, x _i∈ U, 1≤i≤| U|}, wherein, x _iobject in expression field, a represents certain conditional attribute, v _athe value that represents attribute a, the mapping of → representative function, f (x _i, a)=v _aindicated object x _ivalue on attribute a;

If all property values are all known, be called complete Order Strategic Decision system; If there is missing values, be called incomplete Order Strategic Decision system IODS;

Definition 2: suppose x, y ∈ U, b _p(x)={ b|b ∈ P ∧ f (x, b) ≠ * }, puts letter dominance relationship (Confidential Dominance Relation, CDR) and is defined as follows:

CDR(P)={(x,y)∈U ²||B _P(x)∩B _P(y)|/|B _P(x)|=1)∧

Wherein, x, y represents the object in domain, P represents the subset of conditional attribute set C, B _p(x) indicated object x property value is not empty community set, and CDR (P) represents to put the set of letter dominance relationship under property set P, uses

represent " y puts letter advantage in x ";

Definition 3: supposition DS=(U, A, V, f) is an IODS, x ∈ U, the letter advantage of the putting collection of x is: $D_p^{\mathrm{CDR}+}\left(x\right)=\{y\∈U|y{D}_P^{\mathrm{CDR}}x\};$ The letter inferior position collection of putting of x is defined as follows: $D_p^{\mathrm{CDR}-}\left(x\right)=\{y\∈U|x{D}_P^{\mathrm{CDR}}y\};$

Character 1

(1) put letter dominance relationship and meet reflexivity, transitivity and order symmetry;

（2） $G=\left\{D_p^{\mathrm{CDR}+}\left(x\right)\right|x\∈U\}$ And $G=\left\{D_p^{\mathrm{CDR}-}\left(x\right)\right|x\∈U\}$ All coverings of U;

(3) if $x_j\∈D_p^{\mathrm{CDR}+}\left(x_i\right),D_p^{\mathrm{CDR}+}\left(x_j\right)\⊆D_p^{\mathrm{CDR}+}\left(x_i\right);$

If $x_j\∈D_p^{\mathrm{CDR}-}\left(x_i\right),D_p^{\mathrm{CDR}-}\left(x_j\right)\⊆D_p^{\mathrm{CDR}-}\left(x_i\right).$

3. letter dominance based rough set model and the attribute reduction method put according to claim 1, is characterized in that: the definition of described rough set model meets following relational expression:

Suppose that DS=(U, A, V, f) is an IODS, decision attribute D is divided into n class Cl={Cl by domain U _t| t ∈ { 1,2..., n}}, wherein Cl _n> Cl _n-1> ... > Cl ₁, in t class, combine and be defined as under combine and be defined as

show that x at least belongs to class Cl _t,

show that x belongs to class Cl at the most _t; Upper and lower approximate, Boundary Region, approximation quality and nicety of grading based on putting letter dominance relationship are defined as follows:

Definition 4:

x ∈ U,

t=1,2 ..., n,

with

upper and lower be approximately defined as follows respectively:

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\{x\∈U|D_p^{\mathrm{CDR}+}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≥}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≥}}{\∪}{D}_p^{\mathrm{CDR}+}\left(x\right)=\{x|D_p^{\mathrm{CDR}+}\left(x\right)\∩{\mathrm{Cl}}_t^{\≥}\≠\mathrm{\Φ}\};$

$\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\{x\∈U|D_p^{\mathrm{CDR}-}\left(x\right)\⊆{\mathrm{Cl}}_t^{\≤}\},\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)=\underset{x\∈{\mathrm{Cl}}_t^{\≤}}{\∪}{D}_p^{\mathrm{CDR}-}\left(x\right)=\{x|D_p^{\mathrm{CDR}-}\left(x\right)\∩{\mathrm{Cl}}_t^{\≤}\≠\mathrm{\Φ}\};$

Definition 5: Boundary Region is defined as ${\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≥}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right),{\mathrm{Bn}}_p\left({\mathrm{Cl}}_t^{\≤}\right)=\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)-\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right);$

Definition 6: with coarse approximate approximation quality be defined as follows respectively:

$\mathrm{\α}\left({\mathrm{Cl}}_t^{\≥}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≥}\right)\right|,\mathrm{\α}\left({\mathrm{Cl}}_t^{\≤}\right)=\left|\underset{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|/\left|\stackrel{\‾}{P}\left({\mathrm{Cl}}_t^{\≤}\right)\right|;$

Definition: 7: nicety of grading is defined as follows:

${\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≥}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≥}\right)|/\left|U\right|,{\mathrm{\γ}}_P\left({\mathrm{Cl}}^{\≤}\right)=|U-\underset{t=1}{\overset{n}{\∪}}{\mathrm{Bn}}_P\left({\mathrm{Cl}}_t^{\≤}\right)|/\left|U\right|;$

Wherein, Cl represents the set of the classification that decision attribute is divided domain, Cl _trepresent t Decision Classes, t represents classification subscript, in like manner, and Cl _nrepresent n Decision Classes, Cl _n-1represent n-1 Decision Classes, Cl _srepresent s Decision Classes, represent not to be inferior to the object intersection of t Decision Classes,

represent not to be better than the object intersection of t Decision Classes, associating in expression lower approximate set,

the letter advantage of the putting class of indicated object x,

associating in expression

boundary Region,

associating in expression upper approximate set,

represent lower associating lower approximate collection,

represent lower associating

upper approximate collection,

associating in expression

approximation quality,

represent lower associating

approximation quality, γ _p(Cl ^>=) associating in expression

nicety of grading, γ _p(Cl ^≤) the lower associating of expression

nicety of grading, Φ represents empty set.

4. letter dominance based rough set model and the attribute reduction method put according to claim 1, is characterized in that: describedly judge that whether incomplete Order Strategic Decision system IODS is consistent decision system and carries out giving a definition by meeting:

Definition 8: supposition DS=(U, A, V, f) is an IODS, if

claim that decision system DS is consistent; Otherwise, claim that decision system DS is incomparable inconsistent;

Definition 9: supposition decision system DS=(U, A, V, f) is consistent, if and

to arbitrarily

have

claim that P is a yojan of decision system DS;

Definition 10: supposition x, y ∈ U, D*={ (x, y): f (D, x) >=f (D, y) }, the resolution property set of x and y is defined as follows:

the identification matrix that is called decision system DS;

Definition 11: identification function definition is

this function has determined the yojan of incomplete consistent decision system DS;

Wherein, CDR (C) represents to put letter dominance relationship collection under conditional attribute C, and CDR (D) represents to put letter dominance relationship collection under decision attribute D, the value of f (D, x) indicated object x on decision attribute D, f (D, y) represent the value of indicated object y on decision attribute D

indicated object x, the distinctive function of y,

represent identification matrix, △ ^*represent yojan result, ∧ represents conjunction computing, and ∨ represents the computing of extracting.

5. letter dominance based rough set model and the attribute reduction method put according to claim 1, is characterized in that: the described attribute reduction method based on identification matrix comprises the following steps:

S51: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S52: the identification matrix of obtaining decision system DS according to definition 10

S53: according to definition 11 yojan of obtaining about decision system DS.

6. letter dominance based rough set model and the attribute reduction method put according to claim 1, is characterized in that: the described heuristic attribute reduction method based on nicety of grading comprises the following steps:

S61: the letter advantage of the putting class of obtaining each object x ∈ U of decision system DS;

S62: obtain the upper and lower approximate collection of decision system DS, obtain the Boundary Region of decision system DS, calculating nicety of grading is γ _c;

S63: make i=1;

S64:C'=C-{a _i, a wherein _i∈ C;

S65: recalculate C' nicety of grading γ _c';

S66: if γ _c≤ γ _c', remove attribute a _i, C=C', otherwise C'=C;

S67: if i=|C| finishes; Otherwise, i=i+1, execution step 4;

Wherein, U represents field, and x represents domain object, and i represents cycle count, and C represents conditional attribute combination, and C' represents to remove the conditional attribute collection after certain attribute, | C| represents the base of conditional attribute collection, a _irepresent i conditional attribute, γ _cbeing illustrated in conditional attribute integrates as the nicety of grading under C, γ _c'be illustrated in the nicety of grading under the set of C' conditional attribute.

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