CN107301430A - Broad sense Multivariable Fuzzy c means clustering algorithms - Google Patents
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Abstract
The invention discloses a kind of broad sense Multivariable Fuzzy c means clustering algorithms, its step includes：1. pair sample set carries out optimization division according to GMFCM the minimization of object function principle；2. initialization sample component fuzzy membership；3. optimizing estimation is carried out to sample components fuzzy membership using particle cluster algorithm；4. sample components cluster centre is calculated according to component cluster centre formula；5. calculating obtains GMFCM object functions.The present invention assigns fuzzy membership to sample components, enhance classification performance of the clustering algorithm to sample, component fuzzy membership is estimated using particle cluster algorithm, overcome the problem of Multivariable Fuzzy clustering algorithm algorithm is incomplete, fuzzy indicator m is extended to more than to 0 situation simultaneously, the versatility of clustering algorithm is improved.
Description
Technical field
The invention belongs to the algorithm that Data Mining carries out unsupervised data classification, and in particular to one kind is by sample
Component assigns the broad sense Multivariable Fuzzy c means clustering algorithms of fuzzy membership.
Background technology
Fuzzy clustering is an important research content of non-supervised recognition, in machine learning, pattern classification and data
Had a wide range of applications in the fields such as excavation.
FCM Algorithms (fuzzy C-means clustering, FCM) are most important in fuzzy clustering algorithm and should
With one of widest method.There is lot of advantages in FCM algorithms, such as model tormulation formal intuition is readily appreciated, Optimization Solution reason
By it is rigorous, be easy to computer programming is realized, cluster result performance is good etc..
Application and linguistic term on FCM algorithms have a lot, and genetic algorithm (GA) and grain are introduced such as in FCM algorithms
Subgroup (PSO) algorithm optimizes estimation to clustering algorithm parameter, so as to improve FCM algorithm global optimizing abilities.Also have by
FCM algorithms are combined with full variational regularization, and use it for the segmentation that band is made an uproar with fragmentary data image.Multivariable FCM
(multivariable FCM, MFCM) algorithm invests fuzzy membership variable to sample components, to extract more samples
Classification interval information between component, but MFCM algorithms can not handle the situation that sample components are overlapped with cluster centre component.
Fuzzy indicator m is an important parameter in FCM algorithms and its innovatory algorithm, and its proposition directly results in Dunn's
J2Model is extended to the Bezdek clustering function cluster Jm models of updating currently form, and has thus established fuzzy poly- based on object function
The theoretical foundation of class algorithm.Fuzzy indicator m is also referred to as Weighted Index, smoothing parameter, and suitable m values, which have, to be suppressed noise, puts down
The functions such as sliding membership function.
Bedek points out that fuzzy indicator m drastically influence the performance of FCM algorithms, as m=1, and FCM algorithms deteriorate to hard poly-
Class HCM algorithms, when m is intended to just infinite, each class center that FCM algorithms are obtained nearly all is degenerated to the center of gravity of data.
Bedek obtains FCM algorithms fuzzy membership and cluster centre iterative formula using gradient method, in order to ensure FCM convergences of algorithm
Property, it is desirable to thus FCM algorithms object function determines that fuzzy indicator m is necessary to the Second Order Sea plug square formation positive definite of fuzzy membership
More than 1.When solving cluster object function using biological evolution algorithm ambiguous estimation degree of membership, then gradient method second order is avoided
The requirement of sea plug square formation positive definite, so as to which fuzzy indicator m spans are expanded into the situation that m is more than 0.
The content of the invention
Weak point of the invention in order to overcome the presence of MFCM algorithms, it is ensured that clustering algorithm is to sample components and cluster centre
Component overlaps a kind of also effective, broad sense Multivariable Fuzzy c means clustering algorithms (general multivariable of proposition
Fuzzy c-means clustering algorithm), hereinafter referred to as GMFCM, it is therefore intended that biological by introducing population
Evolution algorithm is abandoned so that denominator is zero to cause the gradient method iteration of algorithm contradiction to sample components fuzzy membership optimizing
Formula, so that algorithm is complete effectively.Remain the work(that MFCM algorithms are extracted to sample components class discrimination performance again simultaneously
Can, so as to obtain effective clustering performance, while it is m that also expansible fuzzy indicator, which is expanded,>0 situation, so as to carry
Rise versatility of the clustering algorithm to fuzzy indicator parameter.
In order to realize foregoing invention purpose, the present invention is adopted the following technical scheme that：
The present invention is a kind of broad sense Multivariable Fuzzy c means clustering algorithms, and feature is carried out as follows：
Step 1：Make X={ x1,x2,L,xj,L,xnRepresent given sample set, xjRepresent j-th of sample；1≤j≤n,
N is the number of sample；Optimization division is carried out to sample set X so that target function value JGMFCMMinimum, wherein JGMFCMBy formula
(1) determined.
In formula (1), c represents the classification number divided, 1≤i≤c, j-th of sample1≤j
The corresponding cluster centres of≤n havexjk、θikX is represented respectivelyjAnd θiKth dimension component, 1≤k≤
D, fuzzy membership uijkRepresent sample xjKth dimension component belong to the possibility of the i-th class.0≤uijk≤ 1, and haveRepresent j-th of sample xjKth dimension component belongs to all kinds of fuzzy membership and is 1；U={ uijk,i
=1, L, c；J=1, L, n；K=1, L, d } subordinated-degree matrix is represented, m (m ＞ 0) is fuzzy indicator,For uijkM times；||
xjk-θik| | represent based on j-th of sample xjWith the i-th class cluster centre θiThe two kth tie up the distance of component, generally
It is taken as | xjk-θik|。
Step 2：The position X that multiple c × n × d ties up particle is initialized with the random number between 0,1h (0)With speed Vh (0)。
Step 3：By particle position Xh (0)Component is tieed up as one group using every d, j-th of sample x is corresponded tojKth dimension component be subordinate to
In the fuzzy membership of the i-th classI=1 ..., c, j=1 ..., n, 1≤k≤d, and be normalized so thatDefinition iterations is λ, and maximum iteration is λmax；Initialize λ=1, then the λ times iteration is subordinate to
Matrix is U(λ), the cluster centre of the λ times iteration is θik (λ), cluster centre matrix is P(λ)={ θik (λ), i=1 ..., c, k=
1,...,d}。
Step 4：Cluster centre θ is calculated with formula (2)ik (λ)。
Step 5：PSO algorithm fitness function value f (U are calculated with formula (1) and formula (3)(λ))。
Step 6：Judge | | f (U(λ))-f(U(λ-1)) | | ＜ ε or λ ＞ λmax, if so, then uijk (λ)For iterative algorithm parameter
The optimal fuzzy membership estimated, and make uijk (λ)=uijk, orderAnd then realize to the optimal of sample set X
Divide, ε, λmaxIt is threshold value given in advance.If not, 7 are gone to step, untill condition is met.
Step 7：According to the excellent solution fitness function value f (U of PSO algorithms(λ)), record present age individual in particle cluster algorithm optimal
Solve Ph (λ)With group optimal solution g(λ), λ=λ+1 is made, particle rapidity V is updated by formula (4), (5)h (λ+1)And position Xh (λ+1), turn step
Rapid 3.
Vh (λ+1)=wVh (λ)+c1r1[Ph (λ)-Xh (λ)]+c2r2[g(λ)-Xh (λ)] (4)
Xh (λ+1)=Xh (λ)+Vh (λ+1) (5)
C in formula (4), (5)1, c2For accelerated factor, positive constant is taken as；r1, r2For the random number between [0,1], w claims
For inertial factor.
Compared with the prior art, beneficial effects of the present invention are embodied in：
1.GMFCM algorithms are based on population biological evolution algorithm (PSO) and carry out optimizing to sample components fuzzy membership, no
Dependent on gradient information and component degree of membership iterative formula, it is to avoid denominator has sample components in component degree of membership iterative formula
It is zero situation for causing that algorithm fails with cluster centre distance, so that clustering algorithm is complete effectively.
2.GMFCM algorithms assign fuzzy membership to sample components, can effectively excavate the class discrimination of sample components,
Therefore the clustering performance of clustering algorithm is improved.
3.GMFCM algorithms utilize PSO algorithm sample estimates component fuzzy memberships, make algorithm not by gradient method fuzzy membership
The limitation of Second Order Sea plug square formation positive definite is spent, fuzzy indicator m spans are extended to m ＞ 0 situation, enhance clustering algorithm mould
Paste the universality of index parameter.
Embodiment
In the present embodiment, in order to verify the present invention broad sense Multivariable Fuzzy c means clustering algorithms (hereinafter referred to as
GMFCM algorithms) Cluster Validity, based on dimensional Gaussian data set and UCI data set iris subsets to FCM, GMFCM algorithm
Compare experiment test explanation.When being tested based on MFCM algorithms, iterative algorithm is more when calculating component fuzzy membership
It is secondary occur denominator for 0 so that algorithm failure situation, so emulation experiment has abandoned the contrast test with MFCM algorithms.
Broad sense Multivariable Fuzzy c means clustering algorithms (MFCM) are to carry out as follows：
Step 1：Make X={ x1,x2,L,xj,L,xnRepresent given sample set, xjRepresent j-th of sample；1≤j≤n,
N is the number of sample；Optimization division is carried out to sample set X so that target function value JGMFCMMinimum, wherein JGMFCMBy formula
(1) determined.
1) to do Cluster Validity description of test based on dimensional Gaussian data set as follows.
Construction dimensional Gaussian data set is tested, and clusters classification number C=2, and sample set is that two dimensional Gaussians divide at random
Cloth sample set is constituted, and it is respectively (5,5) to take Liang Leilei centers, (10,10), and the sample number of the first kind is 100, covariance
Matrix is taken as [5 0；0 5], the sample number of Equations of The Second Kind is 100, and covariance matrix is taken as [5 0；0 5].
GMFCM algorithms rely on particle swarm optimization algorithm model, and population uses real coding, and a coding corresponds to
One feasible solution, the positional value of each particle is made up of n × c × d dimensions, and c is classification number, and d is sample dimension, and n is sample
Number.Population is taken as 30, and iterations 200 times, particle often ties up parameter value scope for [0,1], and every c × d of particle position is tieed up
C × d dimension component fuzzy memberships of parameter one sample of correspondence.Clustering Effect pole is absorbed in order to avoid particle group optimizing is calculated
The local optimum of difference, chooses MFCM and trains the sample fuzzy membership come, be configured to an initial grain of particle cluster algorithm
Son, to improve the clustering performance of GMFCM algorithms, that is, has：
uijk(0)=uij * (6)
Formula (6) uijk(0) positional value X during particle cluster algorithm initialization assignment has been corresponded toh(0), uij *Clustered for FCM algorithms
As a result the excellent solution in, relative stability during its object is to using FCM algorithms based on initialization guides GMFCM algorithm iterations
Path.
Test result have recorded all kinds of measuring accuracies, and have recorded the cluster centre of two class data, as shown in table 1.
Test result of the table 1 based on dimensional Gaussian data set
As known from Table 1, it can all be obtained for the hypersphere graphic data with preferable Margin Classification, FCM algorithms and GMFCM algorithms
Preferable classifying quality, classification otherness is little, and GMFCM algorithms utilize PSO algorithm ambiguous estimation degrees of membership, so not by ladder
Degree method Second Order Sea plug battle array positive definite fuzzy indicator m>1 constraint, can be extended to m by fuzzy indicator m spans>0 situation, i.e.,
Can value m=0.2,0.5,0.8 etc., improve universality of the clustering algorithm on fuzzy indicator.
Simultaneously in order to effectively analyze GMFCM algorithms cluster working mechanism, first sample x of data set is taken out1Each component
On all kinds of fuzzy memberships, perform an analysis explanation, as shown in table 2.
First sample components fuzzy membership distribution of the GMFCM algorithms of table 2 based on Gaussian data collection
In table 2, component 1,2 represents sample components 1 and sample components 2, and value corresponding to component k classifications i is to represent
ui1k, u is represented to the total of component fuzzy membershipi1, also it is just sample x1It is attributed to all kinds of degree.Table 2 is given not
With under fuzzy indicator m values, first sample components fuzzy membership value condition, as m=1, GMFCM algorithms are on x1
Classification judgement be hard cluster judgement, that is, have u11+u12=1, u21+u22=0.Each sample components fuzzy membership in table 2
Value, it may be said that bright GMFCM algorithms fuzzy indicator m expansion, overcome MFCM algorithm incompleteness in terms of be it is effective and into
Work(.
2) the validity description of test based on UCI database iris data sets
Test of heuristics, the iris data sets characteristic such as institute of table 3 are carried out based on iris data sets in UCI machine learning databases
Show, test result as shown in table 4, and is given in Table 5 the fuzzy membership distribution of first sample components.
The iris experimental data set attributes of table 3
Test result of the table 4 based on iris data sets
The GMFCM algorithms of table 5 are based on the first sample components fuzzy membership distribution of iris data sets
In formula (1), c represents the classification number divided, 1≤i≤c, j-th of sample1≤j
The corresponding cluster centres of≤n havexjk、θikX is represented respectivelyjAnd θiKth dimension component, 1≤k≤
D, fuzzy membership uijkRepresent sample xjKth dimension component belong to the possibility of the i-th class.0≤uijk≤ 1, and haveRepresent j-th of sample xjKth dimension component belongs to all kinds of fuzzy membership and is 1；U={ uijk,i
=1, L, c；J=1, L, n；K=1, L, d } subordinated-degree matrix is represented, m (m ＞ 0) is fuzzy indicator,For uijkM times；||
xjk-θik| | represent based on j-th of sample xjWith the i-th class cluster centre θiThe two kth tie up the distance of component, generally
It is taken as | xjk-θik|。
Step 2：The position X that multiple c × n × d ties up particle is initialized with the random number between 0,1h (0)With speed Vh (0)。
Step 3：By particle position Xh (0)Component is tieed up as one group using every d, j-th of sample x is corresponded tojKth dimension component be subordinate to
In the fuzzy membership of the i-th classI=1 ..., c, j=1 ..., n, 1≤k≤d, and be normalized so thatDefinition iterations is λ, and maximum iteration is λmax；Initialize λ=1, then the λ times iteration is subordinate to
Matrix is U(λ), the cluster centre of the λ times iteration is θik (λ), cluster centre matrix is P(λ)={ θik (λ), i=1 ..., c, k=
1,...,d}。
Step 4：Cluster centre θ is calculated with formula (2)ik (λ)。
Step 5：PSO algorithm fitness function value f (U are calculated with formula (1) and formula (3)(λ))。
Step 6：Judge | | f (U(λ))-f(U(λ-1)) | | ＜ ε or λ ＞ λ max, if so, then uijk (λ)For iterative algorithm parameter
The optimal fuzzy membership estimated, and make uijk (λ)=uijk, orderAnd then realize to the optimal of sample set X
Divide, ε, λmaxIt is threshold value given in advance.If not, 7 are gone to step, untill condition is met.
Step 7：According to the excellent solution fitness function value f (U of PSO algorithms(λ)), record present age individual in particle cluster algorithm optimal
Solve Ph (λ)With group optimal solution g(λ), λ=λ+1 is made, particle rapidity V is updated by formula (4), (5)h (λ+1)And position Xh (λ+1), turn step
Rapid 3.
Vh (λ+1)=wVh (λ)+c1r1[Ph (λ)-Xh (λ)]+c2r2[g(λ)-Xh (λ)] (4)
Xh (λ+1)=Xh (λ)+Vh (λ+1) (5)
C in formula (4), (5)1, c2For accelerated factor, positive constant is taken as；r1, r2For the random number between [0,1], w claims
For inertial factor.
FCM algorithms are slightly good compared with GMFCM algorithms in the performance that Gaussian data is concentrated, and are worse than GMFCM in iris data sets
Algorithm.Because GMFCM algorithms are, based on the modeling of sample components fuzzy membership, to focus on the distinction of sample components, and it is square
Poor information is an important symbol for showing distinction, so the sample components variance of two data sets is extracted, contrast is ground
The reason for studying carefully two algorithms performance otherness.
The variance of Gaussian data collection two dimensional component is respectively 9.7469,11.6528；The variance of iris data sets four-dimension component
Respectively 0.6857,0.1888,3.1132,0.5824.The otherness of sample components variance is into incrementally becoming in two datasets
Gesture, the performance of GMFCM algorithms is also stepped up.It follows that to be suitable for sample components variance otherness less for FCM algorithms
Data set, and GMFCM algorithms are performed better than in the larger data set of sample components variance otherness.In addition, GMFCM algorithms
Compared with FCM algorithms, due to utilizing particle cluster algorithm sample estimates component fuzzy membership, avoid gradient method ambiguous estimation
Degree of membership is to fuzzy indicator m>1 constraint, can get m>0 scope, enhances the universality and versatility of clustering algorithm.
GMFCM algorithms are because can effectively extract sample components class discrimination, so in the larger number of sample components variance
According to there is preferable performance in collection, it is suitable for the larger unsupervised segmentation problem of high-dimensional weight variance.
In summary, broad sense Multivariable Fuzzy c means clustering algorithms of the invention comprise the following steps：1. pair sample set
Optimization division is carried out according to GMFCM the minimization of object function principle；2. initialization sample component fuzzy membership；3. utilize grain
Swarm optimization carries out optimizing estimation to sample components fuzzy membership；4. sample components are calculated according to component cluster centre formula
Cluster centre；5. calculating obtains GMFCM object functions.The present invention assigns fuzzy membership to sample components, enhances cluster and calculates
Method is estimated component fuzzy membership using particle cluster algorithm the classification performance of sample, overcomes Multivariable Fuzzy and gathers
The problem of class algorithm algorithm is incomplete, while fuzzy indicator m to be extended to more than to 0 situation, improves the general of clustering algorithm
Property.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention.All essences in the present invention
Any modifications, equivalent substitutions and improvements made within refreshing and principle etc., should be included in the scope of the protection.
Claims (6)
1. broad sense Multivariable Fuzzy c means clustering algorithms, comprise the following steps：
Step 1：Sample set is optimized according to broad sense Multivariable Fuzzy c means clustering algorithm the minimization of object function principles
Divide；
Step 2：Initialization sample component fuzzy membership；
Step 3：Optimizing estimation is carried out to sample components fuzzy membership using particle cluster algorithm；
Step 4：Sample components cluster centre is calculated according to component cluster centre formula；
Step 5：Calculating obtains GMFCM object functions.
2. broad sense Multivariable Fuzzy c means clustering algorithms according to claim 1, it is characterised in that：The step 1 is specific
For：Make X={ x1,x2,…,xj,…,xnRepresent given sample set, xjRepresent j-th of sample；1≤j≤n, n are samples
Number；Optimization division is carried out to sample set X so that target function value JGMFCMMinimum, wherein JGMFCMDetermined by formula (1)；
In formula (1), c represents the classification number divided, 1≤i≤c, j-th of sample1≤j≤n phases
The cluster centre answered hasxjk、θikX is represented respectivelyjAnd θiKth dimension component, 1≤k≤d, obscure
Degree of membership uijkRepresent sample xjKth dimension component belong to the possibility of the i-th class.0≤uijk≤ 1, and haveRepresent the
J sample xjKth dimension component belongs to all kinds of fuzzy membership and is 1；U={ uijk, i=1 ..., c；J=1 ..., n；k
=1 ..., d } subordinated-degree matrix is represented, m (m ＞ 0) is fuzzy indicator,For uijkM times；||xjk-θik| | represent to be based on jth
Individual sample xjWith the i-th class cluster centre θiThe two kth tie up the distance of component, be generally taken as | xjk-θik|。
3. broad sense Multivariable Fuzzy c means clustering algorithms according to claim 1, it is characterised in that：The step 2 is specific
For：The position X that multiple c × n × d ties up particle is initialized with the random number between 0,1h (0)With speed Vh (0)。
4. broad sense Multivariable Fuzzy c means clustering algorithms according to claim 1, it is characterised in that：The step 3 is specific
For：By particle position Xh (0)Component is tieed up as one group using every d, j-th of sample x is corresponded tojKth dimension component be under the jurisdiction of the mould of the i-th class
Paste degree of membershipI=1 ..., c, j=1 ..., n, 1≤k≤d, and be normalized so that
Definition iterations is λ, and maximum iteration is λmax；λ=1 is initialized, then the Subject Matrix of the λ times iteration is U(λ), the λ times
The cluster centre of iteration is θik (λ), cluster centre matrix is P(λ)={ θik (λ), i=1 ..., c, k=1 ..., d }.
5. broad sense Multivariable Fuzzy c means clustering algorithms according to claim 1, it is characterised in that：The step 4 is specific
For：Cluster centre θ is calculated with formula (2)ik (λ)；
6. broad sense Multivariable Fuzzy c means clustering algorithms according to claim 2, it is characterised in that：The step 5 is specific
Including：
Step 5-1：PSO algorithm fitness function value f (U are calculated with formula (1) and formula (3)(λ))。
Step 5-2：Judge | | f (U(λ))-f(U(λ-1)) | | ＜ ε or λ ＞ λmax, if so, then uijk (λ)Estimate for iterative algorithm parameter
The optimal fuzzy membership counted out, and make uijk (λ)=uijk, orderAnd then realization is to optimal stroke of sample set X
Point, ε, λmaxIt is threshold value given in advance；If not, 5-3 is gone to step, untill condition is met.
Step 5-3：According to the excellent solution fitness function value f (U of PSO algorithms(λ)), record contemporary individual optimal solution P in particle cluster algorithmh (λ)With group optimal solution g(λ), λ=λ+1 is made, particle rapidity V is updated by formula (4), (5)h (λ+1)And position Xh (λ+1), go to step 3.
Vh (λ+1)=wVh (λ)+c1r1[Ph (λ)-Xh (λ)]+c2r2[g(λ)-Xh (λ)] (4)
Xh (λ+1)=Xh (λ)+Vh (λ+1) (5)
C in formula (4), (5)1, c2For accelerated factor, positive constant is taken as；r1, r2For the random number between [0,1], w is referred to as inertia
The factor.
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