CN107169500A - A kind of Spectral Clustering about subtracted based on neighborhood rough set and system - Google Patents
A kind of Spectral Clustering about subtracted based on neighborhood rough set and system Download PDFInfo
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Abstract
The present invention is a kind of Spectral Clustering about subtracted based on neighborhood rough set and system, and comentropy is incorporated into neighborhood rough set by this method, on the premise of sample separating capacity is kept, and removes the attribute of redundancy, retains and contributes cluster maximum attribute；The attribute set after yojan is then based on, the similarity between sample point, construction similarity matrix and Laplacian Matrix is calculated；Final cluster result is finally obtained using Spectral Clustering.By attribute reduction, the ability that spectral clustering handles high dimensional data can be improved, improves the accuracy rate of cluster.
Description
Technical field
The present invention relates to pattern-recognition and machine learning field, and in particular to a kind of to be gathered based on the spectrum that neighborhood rough set about subtracts
Class method and system.
Background technology
Clustering is an important method of data mining and Information Statistics, and it can be effectively found between things
Inner link, the architectural feature lain in inside data set is depicted to come.The purpose of cluster is sought to according to certain similitude
Measure, if data set is divided into Ganlei's cluster, belongs to of a sort data point with higher similitude, and belongs to different
The data point similitude of class is relatively low.K-means algorithms and FCM algorithms are typical clustering methods, and it is convex spherical that they are adapted to processing
The data set of structure, but for the data set of not convex, algorithm would generally be absorbed in local optimum.
Spectral clustering is clustering algorithm of the class based on graph theory, and clustering problem is changed into Graph partition problem by it, its algorithm frame
Frame generally comprises two big steps：Similarity relation between a similar diagram is constructed first to describe data point；Then according to some
Figure is divided into the data point included in some disconnected subgraphs, subgraph and is considered as class cluster by optimization aim.For the purpose of cluster
Figure to cut optimization aim be usually NP discrete optimization problems, the proposition of spectral clustering allows problem to be asked in polynomial time
Solution.By the Laplacian Matrix of structural map, and to its feature decomposition, then using one or more characteristic vectors, gathered
Class result.
In recent years, advancing by leaps and bounds with science and technology, mass data generates " data explosion ", and these data are usually associated with
Very high dimension, traditional clustering algorithm can not meet the requirement that current data analysis is solved.With traditional clustering algorithm
Compare, spectral clustering has obvious advantage：It can handle increasingly complex clustering architecture (such as non-convex data), and find figure division
The loose solution of the overall situation of object function.Solid theoretical foundation and good Clustering Effect due to having, spectral clustering is expanded should
Use many fields, such as computer vision, IC design, load balancing, biological information, text classification.
But spectral clustering also has the limitation of itself.Some show good spectral clustering in low-dimensional data space, at place
When managing high dimensional data, good cluster result can not be often obtained, or even can fail.Therefore design and develop suitable for higher-dimension magnanimity
The new Spectral Clustering of data analysis, it has also become one of study hotspot both domestic and external.Attribute reduction is one kind of hough transformation
Effective ways, usually as the pre-treatment step of data mining, it is substantially exactly before keeping knowledge-based classification ability constant
Put, delete uncorrelated, unnecessary attribute, so as to reduce the scale of data, computation complexity is reduced, to improve the effect of algorithm
Rate and the precision of data processing.For economic angle, efficient attribute reduction is except that can improve in intelligent information system
Beyond the definition of knowledge, the cost of intelligent information system is also reduced to a certain extent.This embodies " cost minimization,
The business thinking of benefit ", has very important significance for business intelligence.
The content of the invention
In order to solve the above problems, more effectively high dimensional data is clustered, the present invention proposes a kind of based on neighborhood rough set
The Spectral Clustering and system about subtracted.Based on the importance of each attribute of neighborhood rough set theoretical calculation, by Attribute Significance with
Comentropy is combined to select suitable attribute.When the importance of multiple attributes is all identical, just compare the comentropy of attribute, so
The minimum attribute of entropy is added in yojan set afterwards, so as to obtain preferable yojan property set.Followed by the attribute about
Simple method improves spectral clustering, the difference between prominent sample on the basis of keeping sample characteristics so that final is poly-
Class result is divided closer to the real generic of data set.
The present invention is achieved by the following scheme：
The present invention relates to a kind of Spectral Clustering about subtracted based on neighborhood rough set, its principle is：Comentropy is incorporated into
In neighborhood rough set, on the premise of sample separating capacity is kept, the attribute of redundancy is removed, retains and contributes cluster maximum category
Property；The attribute set after yojan is then based on, the similarity between sample point, construction similarity matrix and Laplce's square is calculated
Battle array；Final cluster result is finally obtained using spectral method.This method can weaken noise data and redundant attributes to cluster
Negative effect, with stronger antijamming capability, the operational efficiency and accuracy rate of spectral clustering can be improved to a certain extent.
The related definition of the present invention is as follows：
Define the problem of 1. information system rough sets will be handled and be described as an information system, information system IS=<
U,A,V,f>, wherein U is the nonempty finite set of object, referred to as domain；A is attribute set, is belonged to comprising conditional attribute and decision-making
Property；V is the codomain of all attributes；f:U × A → V is an information function, represents that sample and the correspondence mappings of its attribute value are closed
System.
Define 2. δ-neighborhood domain U={ x1,x2,…,xnIt is a nonempty finite set on real number space, for xi
∈ U, xiδ-neighborhood definition be：
δ(xi)={ x | x ∈ U, Δ (x, xi)≤δ} (1)
Wherein, δ >=0, δ (xi) it is referred to as xiNeighborhood particle, Δ is a distance function.
Define the domain U={ x on the given real number spaces of 3. neighborhood decision system1,x2,…,xn, A represents U attribute
Characteristic set, D represents decision attribute.If A can generate family's neighborhood relationships on domain U, claim NDT=<U,A,D>
It is a neighborhood decision system.
For a neighborhood decision system NDT=<U,A,D>, domain U is divided into N equivalence class by decision attribute D：
X1,X2,…,XN。Decision attribute D is respectively defined as on B lower aprons, the border of upper approximate and decision-making：
Wherein,
Decision attribute D lower apronsNB D is also commonly referred to as the positive domain of decision-making, is denoted as POSB(D)。POSB(D) size reflection
Separable degree of the domain U in given attribute space.Positive domain is bigger, shows that the border of each class is more clear, lap
It is fewer.
This property of 4. attribute dependency according to positive domain is defined, we can define decision attribute D to conditional attribute B
Dependence be：
Wherein, 0≤γB(D)≤1。γB(D) represent in sample set, according to conditional attribute B description, those can be by
The sample that a certain class decision-making is completely included accounts for the ratio of all samples.It is obvious that positive domainNB D is bigger, decision-making D to condition B according to
Lai Xingyue is strong.
The yojan for defining 5. attributes gives a neighborhood decision system NDT=<U,A,D>If B meets following two
Part, then claimIt is an A yojan.
①γB-a(D) ＜ γB(D)；
②γA(D)=γB(D).
1. the condition of definition requires, must can not be independent containing unnecessary attribute, the i.e. yojan in a yojan；Bar
2. part requires that the resolution capability of system should keep constant after yojan.If B1,B2,…,BkIt is system NDT whole yojan,
ThenThe referred to as core of decision system.
Define 6. Attribute Significance and give a neighborhood decision system NDT=<U,A,D>, Then a
Importance relative to B is defined as：
SIG (a, B, D)=γB∪a(D)-γB(D) (6)
Using Attribute Significance index, the old attribute reduction algorithms based on neighborhood rough set can be designed：Calculate first all
The importance of remaining attribute, the attribute for then selecting importance maximum is added in yojan set, repeats said process, Zhi Daosuo
The importance for having remaining attribute is 0, that is, adds any new attribute, the dependence functional value of system all no longer changes.But
It is that sometimes, the maximum attribute of importance may have multiple, traditional Algorithm for Reduction to use the way of optional one, so
Obviously it is more dogmatic, influence of the other factors to Attributions selection is not accounted for, poor yojan result may be caused.Study table
Bright, the attribute reduction from the point of view of information theory can improve the accuracy rate of yojan.
Define 5. given knowledge P and its derived division U/P={ X on domain U1,X2,…,Xn, knowledge P comentropy
It is defined as：
Wherein, p (Xi)=| Xi|/| U | represent equivalence class XiProbability on domain U.
The present invention using comentropy as attribute another evaluation criterion, when the maximum attribute of importance has multiple,
Just compare the size of these Attribute information entropies, the attribute of selection entropy minimum (carrying uncertain information minimum) is incorporated into about
In letter set, so as to obtain the higher yojan property set of the degree of accuracy.
The present invention is comprised the following steps that：
Step 1, for data set X={ x1,x2,…,xn, using the old attribute reduction algorithms based on neighborhood rough set, to sample
This attribute carries out yojan, obtains the property set red. after yojan
Step 1.1：Data set X is considered as neighborhood decision system NDT=<U,A,D>,Calculate neighborhood relationships Na.
Step 1.2：Init attributes set
Step 1.3：Calculate the importance SIG (a of each attributei, red, D) and=γred∪a(D)-γred
(D).
Step 1.4：
IfOnly comprising an attribute, then a is selectedk, meet itOtherwise, the comentropy of these attributes is calculated, a is selectedk, meet it
Step 1.5：If SIG (ak, red, D) and ＞ 0, then akIt is added to yojan concentration, red=red ∪ ak, Ran Houzhuan
To step 1.3；Otherwise, the attribute set red. after output yojan
Step 2, on the basis of yojan property set red, construction similar matrix and Laplacian Matrix, and to Laplce
Matrix- eigenvector-decomposition, calculates its characteristic value and characteristic vector.
Step 2.1：The similarity matrix W ∈ R of data point are set up using the gaussian kernel function of self-regulationn×n, the height of self-regulation
Shown in this kernel function such as formula (8)：
Wherein,d(xi,xj) it is point xiAnd xjEuclidean distance.Formula (8) basis each puts itself
Neighborhood information, be each point xiCalculate an adaptive parameter σi, wherein σiFor point xiTo the average Europe of its pth neighbour
Formula distance.
Step 2.2：Based on similar matrix W, the degree matrix D ∈ R of figure are set up using formula (9)n×n, D is one to angular moment
Battle array：Element on diagonal is di, and off-diagonal element value is 0.
diSummit i degree is represented, is the weights sum with the summit i sides being connected.
Step 2.3：According to similarity matrix W and degree matrix D, construction Laplacian Matrix L=D-1/2(D-W)D-1.
Step 2.4：To matrix L feature decomposition, the corresponding characteristic vector u of its preceding k eigenvalue of maximum is selected1,…,uk,
Then by these characteristic vector longitudinal arrangements, matrix is formed
Step 3, according to the characteristic vector of Laplacian Matrix, the representative point of each Mapping of data points a to low-dimensional,
And these representative points are clustered.
Step 3.1：Standard Process U every a line, unit vector is transformed into by row vector, obtains matrix Y：
Step 3.2：Regard matrix Y every a line as space RkIn a point, utilize k-means or other clustering algorithms
These points are divided into k classes.
Step 3.3：If matrix Y the i-th row is assigned to jth class, just by original data point xiJth class is divided into,
K ready-portioned classes of finally output.
By above content, the application is provided a kind of Spectral Clustering about subtracted based on neighborhood rough set and is
System, comentropy is introduced into neighborhood rough set, as another evaluation criterion of Attribute Significance, when the attribute that importance is maximum
When having multiple, just compare the size of these Attribute information entropies, selection entropy minimum (carrying uncertain information minimum)
Attribute is incorporated into yojan set, so that the higher yojan property set of the degree of accuracy is obtained, then based on the attribute set structure after yojan
Laplacian Matrix is made, and cluster result is obtained using spectral method.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the accompanying drawing used required in technology description to be briefly described, it should be apparent that, drawings in the following description are only this
Some embodiments of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can be with
Other accompanying drawings are obtained according to these accompanying drawings.
A kind of flow chart for spectral clustering about subtracted based on neighborhood rough set that Fig. 1 provides for the embodiment of the present application.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present application, the technical scheme in the embodiment of the present application is carried out clear, complete
Site preparation is described, it is clear that described embodiment is only some embodiments of the present application, rather than whole embodiments.It is based on
Embodiment in the application, it is all other that those of ordinary skill in the art are obtained under the premise of creative work is not paid
Embodiment, belongs to the scope of the application protection.
Embodiment 1
As shown in figure 1, the present embodiment comprises the following steps：
Input：Data set X={ x1,x2,…,xn, clusters number k
Output：K ready-portioned classes
Step 1：According to formula (6) and formula (7), yojan is carried out to data set X attribute, the property set after yojan is obtained
red.
Step 2：On the basis of yojan property set red, the similarity matrix W ∈ R of data point are set up using formula (8)n ×n, the degree matrix D ∈ R of figure are set up using formula (9)n×n.
Step 3：According to similarity matrix W and degree matrix D, construction Laplacian Matrix L=D-1/(2D-W)D-1.
Step 4：To matrix L feature decomposition, the corresponding characteristic vector u of its preceding k eigenvalue of maximum is selected1,…,uk, so
Afterwards by these characteristic vector longitudinal arrangements, matrix is formed
Step 5：Standard Process U every a line, unit vector is transformed into by row vector, obtains matrix Y：
Step 6：Regard matrix Y every a line as space RkIn a point, utilize k-means or other clustering algorithms will
These points are divided into k classes.
Step 7：If matrix Y the i-th row is assigned to jth class, just by original data point xiIt is divided into jth class.
Claims (10)
1. a kind of Spectral Clustering about subtracted based on neighborhood rough set, it is characterised in that by the Importance of attribute in neighborhood rough set
Degree is combined to select suitable attribute with comentropy, on the premise of sample separating capacity is kept, and removes the attribute of redundancy, protects
The attribute maximum to cluster contribution is stayed, the attribute set after yojan is then based on, data point is gathered using Spectral Clustering
Class.
2. according to the method described in claim 1, it is characterized in that, described data set is n × m matrix, matrix it is every
Row represents a data point, and each column represents an attribute, therefore this matrix includes n data point, and each data point has m kinds
Property, X={ x can be expressed as1,x2,…,xn}(xi∈Rm)。
3. according to the method described in claim 1, it is characterized in that, described Attribute Significance refers to：SIG (a, B, D)=γB∪a
(D)-γB(D), wherein D is decision attribute, and B is the subset of conditional attribute, and a is attribute to be analyzed, γB(D) sample set is represented
In conjunction, according to conditional attribute B description, those samples that can be completely included by a certain class decision-making account for the ratio of all samples.
4. according to the method described in claim 1, it is characterized in that, described comentropy refers to：Wherein p (Xi)=| Xi|/| U | represent probability of the equivalence class Xi on domain U.
5. according to the method described in claim 1, it is characterized in that, described yojan refers to：Whole remaining attributes are calculated first
Importance, the attribute for then selecting importance maximum is added in yojan set, said process is repeated, until all remaining attributes
Importance be 0, that is, add any new attribute, the dependence functional value of system all no longer changes.
6. method according to claim 1 or 5, it is characterized in that, described yojan includes：
1：If neighborhood decision system NDT=<U,A,D>, for each attributeCalculate neighborhood relationships Na.
2：Init attributes set
3：Calculate the importance SIG (a of each attributei, red, D) and=γred∪a(D)-γred(D).
4：IfOnly comprising an attribute, then a is selectedk, meet itOtherwise, the comentropy of these attributes is calculated, a is selectedk, meet it
5：If SIG (ak, red, D) and ＞ 0, then akIt is added to yojan concentration, red=red ∪ ak, then proceed to analysis next
Individual attributeOtherwise, the attribute set red after output yojan.
7. according to the method described in claim 1, it is characterized in that, described spectral clustering includes：
1：Similar matrix and Laplacian Matrix are constructed, and to Laplacian Matrix feature decomposition, calculates its characteristic value and feature
Vector.
2：According to the characteristic vector of Laplacian Matrix, the representative point of each Mapping of data points a to low-dimensional, and to these
Point is represented to be clustered.
8. the method according to claim 1 or 7, it is characterized in that, described feature decomposition includes：
1：On the basis of yojan property set red, the gaussian kernel function of self-regulation is utilizedSet up number
The similarity matrix W ∈ R at strong pointn×n；Each data point x is calculated according to matrix WiDegreeBy the degree of n data point
Constitute a diagonal matrix, degree matrix D ∈ Rn×n.
2：According to similarity matrix W and degree matrix D, construction Laplacian Matrix L=D-1/2(D-W)D-1/2.
3：Characteristic vector u corresponding to calculating matrix L preceding k eigenvalue of maximum1,…,uk, then these characteristic vectors are indulged
To arrangement, matrix is formed
4：Standard Process U every a line, unit vector is transformed into by row vector, obtains matrix Y：
9. the method according to claim 1 or 7, it is characterized in that, described cluster includes：
1：Regard matrix Y every a line as space RkIn a point, these points are divided into k using k-means or other algorithms
Class.
2：If matrix Y the i-th row is assigned to jth class, just by original data point xiIt is divided into jth class.
10. a kind of system for realizing any of the above-described claim methods described, it is characterised in that：Attribute reduction module, feature are reflected
Module and cluster module are penetrated, wherein attribute reduction module is according to the importance and comentropy of each attribute in data set, selection pair
The maximum attribute of cluster contribution is added in yojan set, excludes the interference of redundant attributes and noise；Feature Mapping module is based on
Attribute set after yojan, construction similarity matrix and Laplacian Matrix, and the characteristic vector of Laplacian Matrix is calculated,
Mapping of data points is into the low-dimensional feature space being made up of these characteristic vectors；Cluster module is using k-means algorithms to feature
Representative point in space is clustered, and exports cluster result.
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| CN108345720A (en) * | 2018-01-18 | 2018-07-31 | 河海大学 | Dam health status influence factor contribution degree discrimination method in a kind of full-time spatial domain |
| CN109447972A (en) * | 2018-10-31 | 2019-03-08 | 岭南师范学院 | A kind of high spectrum image discrimination method detecting soybean thermal damage |
| CN110276393A (en) * | 2019-06-19 | 2019-09-24 | 西安建筑科技大学 | A kind of compound prediction technique of green building energy consumption |
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Application publication date: 20170915 |