Fuzzy adjusted Rand index

Description

Produces the fuzzy version of the adjusted Rand index between a hard (reference) partition and a fuzzy partition.

Usage

ARI.F(VC, U, t_norm)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Campello, R.J., 2007. A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognition Letters, 28, 833-841.
Hubert, L., Arabie, P., 1985. Comparing partitions. Journal of Classification, 2, 193-218.

See Also

RI.F, JACCARD.F, Fclust.compare

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## fuzzy adjusted Rand index
ari.f=ARI.F(VC=Mc$Type,U=clust$U)

## End(Not run)

Fuzzy k-means

Description

Performs the fuzzy k-means clustering algorithm.

Usage

 
 FKM (X, k, m, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York.

See Also

FKM.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means (excluded the factor column Type (last column)), fixing the number of clusters
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## fuzzy k-means (excluded the factor column Type (last column)), selecting the number of clusters
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=2:6,m=1.5,stand=1)

Fuzzy k-means with entropy regularization

Description

Performs the fuzzy k-means clustering algorithm with entropy regularization.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).

Usage

 
 FKM.ent (X, k, ent, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
The default value for ent is in general not feasible if FKM.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.ent using standardized data (stand=1).

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Li R., Mukaidono M., 1995. A maximum entropy approach to fuzzy clustering. Proceedings of the Fourth IEEE Conference on Fuzzy Systems (FUZZ-IEEE/IFES '95), pp. 2227-2232.
Li R., Mukaidono M., 1999. Gaussian clustering method based on maximum-fuzzy-entropy interpretation. Fuzzy Sets and Systems, 102, 253-258.

See Also

FKM.ent.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means with entropy regularization, fixing the number of clusters
## (excluded the factor column Type (last column))
clust=FKM.ent(Mc[,1:(ncol(Mc)-1)],k=6,ent=3,RS=10,stand=1)
## fuzzy k-means with entropy regularization, selecting the number of clusters
## (excluded the factor column Type (last column))
clust=FKM.ent(Mc[,1:(ncol(Mc)-1)],k=2:6,ent=3,RS=10,stand=1)

Fuzzy k-means with entropy regularization and noise cluster

Description

Performs the fuzzy k-means clustering algorithm with entropy regularization and noise cluster.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 
 FKM.ent.noise (X, k, ent, delta, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
The default value for ent is in general not feasible if FKM.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.ent using standardized data (stand=1).

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Li R., Mukaidono M., 1995. A maximum entropy approach to fuzzy clustering. Proceedings of the Fourth IEEE Conference on Fuzzy Systems (FUZZ-IEEE/IFES '95), pp. 2227-2232.
Li R., Mukaidono M., 1999. Gaussian clustering method based on maximum-fuzzy-entropy interpretation. Fuzzy Sets and Systems, 102, 253-258.

See Also

FKM.ent, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, butterfly

Examples

## butterfly data
data(butterfly)
## fuzzy k-means with entropy regularization and noise cluster, fixing the number of clusters
clust=FKM.ent.noise(butterfly,k = 2, RS=5,delta=3)
## fuzzy k-means with entropy regularization and noise cluster, selecting the number of clusters
clust=FKM.ent.noise(butterfly,RS=5,delta=3)

Gustafson and Kessel - like fuzzy k-means

Description

Performs the Gustafson and Kessel - like fuzzy k-means clustering algorithm.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.

Usage

 FKM.gk (X, k, m, vp, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, then the algorithm stops and the element of value is NaN.
The Babuska et al. variant in FKM.gkb is recommended.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

FKM.gkb, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson and Kessel-like fuzzy k-means, fixing the number of clusters
clust=FKM.gk(unemployment,k=3,RS=10)
## Gustafson and Kessel-like fuzzy k-means, selecting the number of clusters
clust=FKM.gk(unemployment,k=2:6,RS=10)

## End(Not run)

Gustafson and Kessel - like fuzzy k-means with entropy regularization

Description

Performs the Gustafson and Kessel - like fuzzy k-means clustering algorithm with entropy regularization.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).

Usage

 
 FKM.gk.ent (X, k, ent, vp, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, the algorithm stops and the element of value is NaN.
The default value for ent is in general not reasonable if FKM.gk.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.gk.ent using standardized data (stand=1).
The Babuska et al. variant in FKM.gkb.ent is recommended.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Ferraro M.B., Giordani P., 2013. A new fuzzy clustering algorithm with entropy regularization. Proceedings of the meeting on Classification and Data Analysis (CLADAG).

See Also

FKM.gkb.ent, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## unemployment data
data(unemployment)
## Gustafson and Kessel-like fuzzy k-means with entropy regularization, 
##fixing the number of clusters
clust=FKM.gk.ent(unemployment,k=3,ent=0.2,RS=10,stand=1)
## Not run: 
## Gustafson and Kessel-like fuzzy k-means with entropy regularization,
##selecting the number of clusters
clust=FKM.gk.ent(unemployment,k=2:6,ent=0.2,RS=10,stand=1)

## End(Not run)

Gustafson and Kessel - like fuzzy k-means with entropy regularization and noise cluster

Description

Performs the Gustafson and Kessel - like fuzzy k-means clustering algorithm with entropy regularization and noise cluster.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 FKM.gk.ent.noise (X,k,ent,vp,delta,RS,stand,startU,index,alpha,conv,maxit,seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, the algorithm stops and the element of value is NaN.
The default value for ent is in general not feasible if FKM.gk.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.gk.ent.noise using standardized data (stand=1).
The Babuska et al. variant in FKM.gkb.ent.noise is recommended.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Ferraro M.B., Giordani P., 2013. A new fuzzy clustering algorithm with entropy regularization. Proceedings of the meeting on Classification and Data Analysis (CLADAG).

See Also

FKM.gkb.ent.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson and Kessel-like fuzzy k-means with entropy regularization and noise cluster,
##fixing the number of clusters
clust=FKM.gk.ent.noise(unemployment,k=3,ent=0.2,delta=1,RS=10,stand=1)
## Gustafson and Kessel-like fuzzy k-means with entropy regularization and noise cluster,
##selecting the number of clusters
clust=FKM.gk.ent.noise(unemployment,k=2:6,ent=0.2,delta=1,RS=10,stand=1)

## End(Not run)

Gustafson and Kessel - like fuzzy k-means with noise cluster

Description

Performs the Gustafson and Kessel - like fuzzy k-means clustering algorithm with noise cluster.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 FKM.gk.noise (X, k, m, vp, delta, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, then the algorithm stops and the element of value is NaN.
The Babuska et al. variant in FKM.gkb.noise is recommended.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

FKM.gkb.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson and Kessel-like fuzzy k-means with noise cluster, fixing the number of clusters
clust=FKM.gk.noise(unemployment,k=3,delta=20,RS=10)
## Gustafson and Kessel-like fuzzy k-means with noise cluster, selecting the number of clusters
clust=FKM.gk.noise(unemployment,k=2:6,delta=20,RS=10)

## End(Not run)

Gustafson, Kessel and Babuska - like fuzzy k-means

Description

Performs the Gustafson, Kessel and Babuska - like fuzzy k-means clustering algorithm.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The Babuska et al. variant improves the computation of the fuzzy covariance matrices in the standard Gustafson and Kessel clustering algorithm.

Usage

 FKM.gkb (X, k, m, vp, gam, mcn, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, then the algorithm stops and the element of value is NaN.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Babuska R., van der Veen P.J., Kaymak U., 2002. Improved covariance estimation for Gustafson-Kessel clustering. Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 1081-1085.
Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

FKM.gk, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson, Kessel and Babuska-like fuzzy k-means, fixing the number of clusters
clust=FKM.gkb(unemployment,k=3,RS=10)
## Gustafson, Kessel and Babuska-like fuzzy k-means, selecting the number of clusters
clust=FKM.gkb(unemployment,k=2:6,RS=10)
## End(Not run)

Gustafson, Kessel and Babuska - like fuzzy k-means with entropy regularization

Description

Performs the Gustafson, Kessel and Babuska - like fuzzy k-means clustering algorithm with entropy regularization.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The Babuska et al. variant improves the computation of the fuzzy covariance matrices in the standard Gustafson and Kessel clustering algorithm.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).

Usage

 FKM.gkb.ent (X, k, ent, vp, gam, mcn, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, the algorithm stops and the element of value is NaN.
The default value for ent is in general not reasonable if FKM.gk.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.gk.ent using standardized data (stand=1).

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Babuska R., van der Veen P.J., Kaymak U., 2002. Improved covariance estimation for Gustafson-Kessel clustering. Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 1081-1085.
Ferraro M.B., Giordani P., 2013. A new fuzzy clustering algorithm with entropy regularization. Proceedings of the meeting on Classification and Data Analysis (CLADAG).

See Also

FKM.gk.ent, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson, Kessel and Babuska-like fuzzy k-means with entropy regularization,
##fixing the number of clusters
clust=FKM.gkb.ent(unemployment,k=3,ent=0.2,RS=10,stand=1)
## Gustafson, Kessel and Babuska-like fuzzy k-means with entropy regularization,
##selecting the number of clusters
clust=FKM.gkb.ent(unemployment,k=2:6,ent=0.2,RS=10,stand=1)
## End(Not run)

Gustafson, Kessel and Babuska - like fuzzy k-means with entropy regularization and noise cluster

Description

Performs the Gustafson, Kessel and Babuska - like fuzzy k-means clustering algorithm with entropy regularization and noise cluster.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The Babuska et al. variant improves the computation of the fuzzy covariance matrices in the standard Gustafson and Kessel clustering algorithm.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 FKM.gkb.ent.noise (X,k,ent,vp,delta,gam,mcn,RS,stand,startU,index,alpha,conv,maxit,seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, the algorithm stops and the element of value is NaN.
The default value for ent is in general not reasonable if FKM.gk.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.gk.ent.noise using standardized data (stand=1).

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Babuska R., van der Veen P.J., Kaymak U., 2002. Improved covariance estimation for Gustafson-Kessel clustering. Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 1081-1085.
Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Ferraro M.B., Giordani P., 2013. A new fuzzy clustering algorithm with entropy regularization. Proceedings of the meeting on Classification and Data Analysis (CLADAG).

See Also

FKM.gk.ent.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson, Kessel and Babuska-like fuzzy k-means with entropy regularization and noise cluster,
##fixing the number of clusters
clust=FKM.gkb.ent.noise(unemployment,k=3,ent=0.2,delta=1,RS=10,stand=1)
## Gustafson, Kessel and Babuska-like fuzzy k-means with entropy regularization and noise cluster,
##selecting the number of clusters
clust=FKM.gkb.ent.noise(unemployment,k=2:6,ent=0.2,delta=1,RS=10,stand=1)

## End(Not run)

Gustafson, Kessel and Babuska - like fuzzy k-means with noise cluster

Description

Performs the Gustafson, Kessel and Babuska - like fuzzy k-means clustering algorithm with noise cluster.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.
The Babuska et al. variant improves the computation of the fuzzy covariance matrices in the standard Gustafson and Kessel clustering algorithm.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 FKM.gkb.noise (X,k,m,vp,delta,gam,mcn,RS,stand,startU,index,alpha,conv,maxit,seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, then the algorithm stops and the element of value is NaN.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Babuska R., van der Veen P.J., Kaymak U., 2002. Improved covariance estimation for Gustafson-Kessel clustering. Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 1081-1085.
Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

FKM.gk.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson, Kessel and Babuska-like fuzzy k-means with noise cluster,
##fixing the number of clusters
clust=FKM.gkb.noise(unemployment,k=3,delta=20,RS=10)
## Gustafson, Kessel and Babuska-like fuzzy k-means with noise cluster,
##selecting the number of clusters
clust=FKM.gkb.noise(unemployment,k=2:6,delta=20,RS=10)
## End(Not run)

Fuzzy k-medoids

Description

Performs the fuzzy k-medoids clustering algorithm.
Differently from fuzzy k-means where the cluster prototypes (centroids) are artificial objects computed as weighted means, in the fuzzy k-medoids the cluster prototypes (medoids) are a subset of the observed objects.

Usage

 FKM.med (X, k, m, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
In FKM.med the parameter of fuzziness is usually lower than the one used in FKM.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Krishnapuram R., Joshi A., Nasraoui O., Yi L., 2001. Low-complexity fuzzy relational clustering algorithms for web mining. IEEE Transactions on Fuzzy Systems, 9, 595-607.

See Also

FKM.med.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, Mc

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-medoids, fixing the number of clusters
## (excluded the factor column Type (last column))
clust=FKM.med(Mc[,1:(ncol(Mc)-1)],k=6,m=1.1,RS=10,stand=1)
## fuzzy k-medoids, selecting the number of clusters
## (excluded the factor column Type (last column))
clust=FKM.med(Mc[,1:(ncol(Mc)-1)],k=2:6,m=1.1,RS=10,stand=1)

## End(Not run)

Fuzzy k-medoids with noise cluster

Description

Performs the fuzzy k-medoids clustering algorithm with noise cluster.
Differently from fuzzy k-means where the cluster prototypes (centroids) are artificial objects computed as weighted means, in the fuzzy k-medoids the cluster prototypes (medoids) are a subset of the observed objects.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 
 FKM.med.noise (X, k, m, delta, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
As for FKM.med, in FKM.med.noise the parameter of fuzziness is usually lower than the one used in FKM.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Krishnapuram R., Joshi A., Nasraoui O., Yi L., 2001. Low-complexity fuzzy relational clustering algorithms for web mining. IEEE Transactions on Fuzzy Systems, 9, 595-607.

See Also

FKM.med, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, butterfly

Examples

## butterfly data
data(butterfly)
## fuzzy k-medoids with noise cluster, fixing the number of clusters
clust=FKM.med.noise(butterfly,k=2,RS=5,delta=3)
## fuzzy k-medoids with noise cluster, selecting the number of clusters
clust=FKM.med.noise(butterfly,RS=5,delta=3)

Fuzzy k-means with noise cluster

Description

Performs the fuzzy k-means clustering algorithm with noise cluster.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 
 FKM.noise (X, k, m, delta, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.

See Also

FKM, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, butterfly

Examples

## butterfly data
data(butterfly)
## fuzzy k-means with noise cluster, fixing the number of clusters
clust=FKM.noise(butterfly, k = 2, RS=5,delta=3)
## fuzzy k-means with noise cluster, selecting the number of clusters
clust=FKM.noise(butterfly,RS=5,delta=3)

Fuzzy k-means with polynomial fuzzifier

Description

Performs the fuzzy k-means clustering algorithm with polynomial fuzzifier function.
The polynomial fuzzifier creates areas of crisp membership degrees around the prototypes while, outside of these areas of crisp membership degrees, fuzzy membership degrees are given. Therefore, the polynomial fuzzifier produces membership degrees equal to one for objects clearly assigned to clusters, that is, very close to the cluster prototypes.

Usage

 
 FKM.pf (X, k, b, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Winkler R., Klawonn F., Hoeppner F., Kruse R., 2010. Fuzzy Cluster Analysis of Larger Data Sets. In: Scalable Fuzzy Algorithms for Data Management and Analysis: Methods and Design IGI Global, pp. 302-331. IGI Global, Hershey.
Winkler R., Klawonn F., Kruse R., 2011. Fuzzy clustering with polynomial fuzzifier function in connection with M-estimators. Applied and Computational Mathematics, 10, 146-163.

See Also

FKM.pf.noise, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, Mc

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means with polynomial fuzzifier, fixing the number of clusters 
## (excluded the factor column Type (last column))
clust=FKM.pf(Mc[,1:(ncol(Mc)-1)],k=6,stand=1)
## fuzzy k-means with polynomial fuzzifier, selecting the number of clusters  
## (excluded the factor column Type (last column))
clust=FKM.pf(Mc[,1:(ncol(Mc)-1)],k=2:6,stand=1)
## End(Not run)

Fuzzy k-means with polynomial fuzzifier and noise cluster

Description

Performs the fuzzy k-means clustering algorithm with polynomial fuzzifier function and noise cluster.
The polynomial fuzzifier creates areas of crisp membership degrees around the prototypes while, outside of these areas of crisp membership degrees, fuzzy membership degrees are given. Therefore, the polynomial fuzzifier produces membership degrees equal to one for objects clearly assigned to clusters, that is, very close to the cluster prototypes.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 
 FKM.pf.noise (X, k, b, delta, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Winkler R., Klawonn F., Hoeppner F., Kruse R., 2010. Fuzzy cluster analysis of larger data sets. In: Scalable Fuzzy Algorithms for Data Management and Analysis: Methods and Design IGI Global, pp. 302-331. IGI Global, Hershey.
Winkler R., Klawonn F., Kruse R., 2011. Fuzzy clustering with polynomial fuzzifier function in connection with M-estimators. Applied and Computational Mathematics, 10, 146-163.

See Also

FKM.pf, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, Mc

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means with polynomial fuzzifier and noise cluster, fixing the number of clusters 
## (excluded the factor column Type (last column))
clust=FKM.pf.noise(Mc[,1:(ncol(Mc)-1)],k=6,stand=1)
## fuzzy k-means with polynomial fuzzifier and noise cluster, selecting the number of clusters 
## (excluded the factor column Type (last column))
clust=FKM.pf.noise(Mc[,1:(ncol(Mc)-1)],k=2:6,stand=1)
## End(Not run)

Fuzzy clustering

Description

Performs fuzzy clustering by using the algorithms available in the package.

Usage

 Fclust (X, k, type, ent, noise, stand, distance)

Arguments

Details

The clustering algorithms are run by using default options.
To specify different options, use the corresponding function.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

print.fclust, summary.fclust, plot.fclust, FKM, FKM.ent, FKM.gk, FKM.gk.ent, FKM.gkb, FKM.gkb.ent, FKM.med, FKM.pf, FKM.noise, FKM.ent.noise, FKM.gk.noise, FKM.gkb.ent.noise, FKM.gkb.noise, FKM.gk.ent.noise,FKM.med.noise, FKM.pf.noise, NEFRC, NEFRC.noise, Fclust.index, Fclust.compare

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=Fclust(Mc[,1:(ncol(Mc)-1)],k=6,type="standard",ent=FALSE,noise=FALSE,stand=1,distance=FALSE)
## fuzzy k-means with polynomial fuzzifier 
## (excluded the factor column Type (last column))
clust=Fclust(Mc[,1:(ncol(Mc)-1)],k=6,type="polynomial",ent=FALSE,noise=FALSE,stand=1,distance=FALSE)
## fuzzy k-means with entropy regularization
## (excluded the factor column Type (last column))
clust=Fclust(Mc[,1:(ncol(Mc)-1)],k=6,type="standard",ent=TRUE,noise=FALSE,stand=1,distance=FALSE)
## fuzzy k-means with noise cluster
## (excluded the factor column Type (last column))
clust=Fclust(Mc[,1:(ncol(Mc)-1)],k=6,type="standard",ent=FALSE,noise=TRUE,stand=1,distance=FALSE)

## End(Not run)

Similarity between partitions

Description

Performs some measures of similarity between a hard (reference) partition and a fuzzy partition.

Usage

Fclust.compare(VC, U, index, tnorm)

Arguments

Details

index is not case-sensitive. All the measures of similarity share the same properties of their non-fuzzy counterpart.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Campello, R.J., 2007. A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognition Letters, 28, 833-841.
Hubert, L., Arabie, P., 1985. Comparing partitions. Journal of Classification, 2, 193-218.
Jaccard, P., 1901. Étude comparative de la distribution florale dans une portion des Alpes et des Jura. Bulletin de la Société Vaudoise des Sciences Naturelles, 37, 547-579.
Rand, W.M., 1971. Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846-850.

See Also

RI.F, ARI.F, JACCARD.F

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## all measures of similarity
all.indexes=Fclust.compare(VC=Mc$Type,U=clust$U)
## fuzzy adjusted Rand index
Fari.index=Fclust.compare(VC=Mc$Type,U=clust$U,index="ARI.F")

## End(Not run)

Cluster validity indexes

Description

Performs some cluster validity indexes for choosing the optimal number of clusters k.

Usage

 Fclust.index (fclust.obj, index, alpha)

Arguments

Details

index is not case-sensitive.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

PC, PE, MPC, SIL, SIL.F, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## cluster validity indexes
all.indexes=Fclust.index(clust)
## Xie and Beni cluster validity index
XB.index=Fclust.index(clust,'XB')

Raw prototypes

Description

Produces prototypes using the original units of measurement of X (useful if the clustering algorithm is run using standardized data).

Usage

 Hraw (X, H)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, unemployment

Examples

## example n.1 (k-means case)
## unemployment data
data(unemployment)
## fuzzy k-means
unempFKM=FKM(unemployment,k=3,stand=1)
## standardized prototypes
unempFKM$H
## prototypes using the original units of measurement
unempFKM$Hraw=Hraw(unempFKM$X,unempFKM$H)
## example n.2  (k-medoids case)
## unemployment data
data(unemployment)
## fuzzy k-medoids
## Not run: 
## It may take more than a few seconds
unempFKM.med=FKM.med(unemployment,k=3,RS=10,stand=1)
## prototypes using the original units of measurement:
## in fuzzy k-medoids one can equivalently use
unempFKM.med$Hraw1=Hraw(unempFKM.med$X,unempFKM.med$H)
unempFKM.med$Hraw2=unempFKM.med$X[unempFKM.med$medoid,]
## End(Not run)

Fuzzy Jaccard index

Description

Produces the fuzzy version of the Jaccard index between a hard (reference) partition and a fuzzy partition.

Usage

JACCARD.F(VC, U, t_norm)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Campello, R.J., 2007. A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognition Letters, 28, 833-841.
Jaccard, P., 1901. Étude comparative de la distribution florale dans une portion des Alpes et des Jura. Bulletin de la Société Vaudoise des Sciences Naturelles, 37, 547-579.

See Also

ARI.F, RI.F, Fclust.compare

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## fuzzy Jaccard index
jaccard.f=JACCARD.F(VC=Mc$Type,U=clust$U)

## End(Not run)

Modified partition coefficient

Description

Produces the modified partition coefficient index. The optimal number of clusters k is such that the index takes the maximum value.

Usage

 MPC (U)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1996. Validating fuzzy partitions obtained through c-shells clustering. Pattern Recognition Letters, 17, 613-623.

See Also

PC, PE, SIL, SIL.F, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## modified partition coefficient
mpc=MPC(clust$U)

McDonald's data

Description

Nutrition analysis of McDonald's menu items.

Usage

data(Mc)

Format

A data.frame with 81 rows and 16 columns.

Details

Data are from McDonald's USA Nutrition Facts for Popular Menu Items. A subset of menu items is reported. Beverages are excluded. In case of duplications, regular size or medium size information is reported. The variable Type is a factor the levels of which specify the kind of the menu items. Although some menu items could be well described by more than one level, only one level of the variable Type specifies each menu item. Percent Daily Values (%DV) are based on a 2,000 calorie diet. Some menu items are registered trademarks.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, FKM, FKM.ent, FKM.med

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
p=(ncol(Mc)-1)
## fuzzy k-means (excluded the factor column Type (last column))
clust.FKM=FKM(Mc[,1:p],k=6,m=1.5,stand=1)
## new factor column Cluster.FKM containing the cluster assignment information
## using fuzzy k-means
Mc[,ncol(Mc)+1]=factor(clust.FKM$clus[,1])
colnames(Mc)[ncol(Mc)]=("Cluster.FKM")
levels(Mc$Cluster.FKM)=paste("Clus FKM",1:clust.FKM$k,sep=" ")
## contingency table (Cluster.FKM vs Type)
## to assess whether clusters can be interpreted in terms of the levels of Type
table(Mc$Type,Mc$Cluster.FKM)
## prototypes using the original units of measurement
clust.FKM$Hraw=Hraw(clust.FKM$X,clust.FKM$H)
clust.FKM$Hraw
## fuzzy k-means with entropy regularization
## (excluded the factor column Type (last column))
## Not run: 
## It may take more than a few seconds
clust.FKM.ent=FKM.ent(Mc[,1:p],k=6,ent=3,RS=10,stand=1)
## new factor column Cluster.FKM.ent containing the cluster assignment information
## using fuzzy k-medoids with entropy regularization
Mc[,ncol(Mc)+1]=factor(clust.FKM.ent$clus[,1])
colnames(Mc)[ncol(Mc)]=("Cluster.FKM.ent")
levels(Mc$Cluster.FKM.ent)=paste("Clus FKM.ent",1:clust.FKM.ent$k,sep=" ")
## contingency table (Cluster.FKM.ent vs Type)
## to assess whether clusters can be interpreted in terms of the levels of Type
table(Mc$Type,Mc$Cluster.FKM.ent)
## prototypes using the original units of measurement
clust.FKM.ent$Hraw=Hraw(clust.FKM.ent$X,clust.FKM.ent$H)
clust.FKM.ent$Hraw
## End(Not run)
## fuzzy k-medoids
## (excluded the factor column Type (last column))
clust.FKM.med=FKM.med(Mc[,1:p],k=6,m=1.1,RS=10,stand=1)
## new factor column Cluster.FKM.med containing the cluster assignment information
## using fuzzy k-medoids with entropy regularization
Mc[,ncol(Mc)+1]=factor(clust.FKM.med$clus[,1])
colnames(Mc)[ncol(Mc)]=("Cluster.FKM.med")
levels(Mc$Cluster.FKM.med)=paste("Clus FKM.med",1:clust.FKM.med$k,sep=" ")
## contingency table (Cluster.FKM.med vs Type)
## to assess whether clusters can be interpreted in terms of the levels of Type
table(Mc$Type,Mc$Cluster.FKM.med)
## prototypes using the original units of measurement
clust.FKM.med$Hraw=Hraw(clust.FKM.med$X,clust.FKM.med$H)
clust.FKM.med$Hraw
## or, equivalently,
Mc[clust.FKM.med$medoid,1:p]

NBA teams data

Description

NBA team statistics from the 2017-2018 regular season.

Usage

data(NBA)

Format

A data.frame with 30 rows and 22 columns.

Details

Data refer to some statistics of the NBA teams for the regular season 2017-2018. The teams are distinguished according to two classification variables.

The statistics are: number of wins (W), field goals made (FGM), field goals attempted (FGA), field goals percentage (FGP), 3 point field goals made (3PM), 3 point field goals attempted (3PA), 3 point field goals percentage (3PP), free throws made (FTM), free throws attempted (FTA), free throws percentage (FTP), offensive rebounds (OREB), defensive rebounds (DREB), assists (AST), turnovers (TOV), steals (STL), blocks (BLK), blocked field goal attempts (BLKA), personal fouls (PF), personal fouls drawn (PFD) and points (PTS). Moreover, reported are the conference (Conference) and the playoff appearance (Playoff).

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

Source

https://stats.nba.com/teams/traditional/

See Also

FKM

Examples

## Not run: 

data(NBA)
## A subset of variables is considered
X <- NBA[,c(4,7,10,11,12,13,14,15,16,17,20)]
clust.FKM=FKM(X=X,k=2:6,m=1.5,RS=50,stand=1,index="SIL.F",alpha=1)
summary(clust.FKM)

## End(Not run)

Non-Euclidean Fuzzy Relational Clustering

Description

Performs the Non-Euclidean Fuzzy Relational data Clustering algorithm.

Usage

NEFRC(D, k, m, RS, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Davé, R. N., & Sen, S. 2002. Robust fuzzy clustering of relational data. IEEE Transactions on Fuzzy Systems, 10(6), 713-727.

See Also

NEFRC.noise, print.fclust, summary.fclust, plot.fclust

Examples

## Not run: 
require(cluster)
data("houseVotes")
X <- houseVotes[,-1]
D <- daisy(x = X, metric = "gower")
clust.NEFRC <- NEFRC(D = D, k = 2:6, m = 2, index = "SIL.F")
summary(clust.NEFRC)
plot(clust.NEFRC)

## End(Not run)

Non-Euclidean Fuzzy Relational Clustering with noise cluster

Description

Performs the Non-Euclidean Fuzzy Relational data Clustering algorithm.
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

NEFRC.noise(D, k, m, delta, RS, startU, index, alpha, conv, maxit, seed)

Arguments

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.

Value

Object of class fclust, which is a list with the following components:

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Davé, R. N., & Sen, S. 2002. Robust fuzzy clustering of relational data. IEEE Transactions on Fuzzy Systems, 10(6), 713-727.

See Also

NEFRC, print.fclust, summary.fclust, plot.fclust

Examples

## Not run: 
require(cluster)
data("houseVotes")
X <- houseVotes[,-1]
D <- daisy(x = X, metric = "gower")
clust.NEFRC.noise <- NEFRC.noise(D = D, k = 2:6, m = 2, index = "SIL.F")
summary(clust.NEFRC.noise)
plot(clust.NEFRC.noise)

## End(Not run)

Partition coefficient

Description

Produces the partition coefficient index. The optimal number of clusters k is is such that the index takes the maximum value.

Usage

 PC (U)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., 1974. Cluster validity with fuzzy sets. Journal of Cybernetics, 3, 58-73.

See Also

PE, MPC, SIL, SIL.F, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## partition coefficient
pc=PC(clust$U)

Partition entropy

Description

Produces the partition entropy index. The optimal number of clusters k is is such that the index takes the minimum value.

Usage

 PE (U, b)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York.

See Also

PC, MPC, SIL, SIL.F, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## partition entropy index
pe=PE(clust$U)

Fuzzy Rand index

Description

Produces the fuzzy version of the Rand index between a hard (reference) partition and a fuzzy partition.

Usage

RI.F(VC, U, t_norm)

Arguments

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Campello, R.J., 2007. A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognition Letters, 28, 833-841.
Rand, W.M., 1971. Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846-850.

See Also

ARI.F, JACCARD.F, Fclust.compare

Examples

## Not run: 
## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## fuzzy Rand index
ri.f=RI.F(VC=Mc$Type,U=clust$U)

## End(Not run)

Silhouette index

Description

Produces the silhouette index. The optimal number of clusters k is is such that the index takes the maximum value.

Usage

 SIL (Xca, U, distance)

Arguments

Details

Xca should contain the same dataset used in the clustering algorithm, i.e., if the clustering algorithm is run using standardized data, then SIL should be computed using the same standardized data.
Set distance=TRUE if Xca is a distance/dissimilarity matrix.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Kaufman L., Rousseeuw P.J., 1990. Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.

See Also

PC, PE, MPC, SIL.F, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## silhouette index
sil=SIL(clust$Xca,clust$U)

Fuzzy silhouette index

Description

Produces the fuzzy silhouette index. The optimal number of clusters k is is such that the index takes the maximum value.

Usage

 SIL.F (Xca, U, alpha, distance)

Arguments

Details

Xca should contain the same dataset used in the clustering algorithm, i.e., if the clustering algorithm is run using standardized data, then SIL.F should be computed using the same standardized data.
Set distance=TRUE if Xca is a distance/dissimilarity matrix.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Campello R.J.G.B., Hruschka E.R., 2006. A fuzzy extension of the silhouette width criterion for cluster analysis. Fuzzy Sets and Systems, 157, 2858-2875.

See Also

PC, PE, MPC, SIL, XB, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## fuzzy silhouette index
sil.f=SIL.F(clust$Xca,clust$U)

Visual Assessment of (Cluster) Tendency

Description

Digital intensity image to inspect the number of clusters

Usage

 VAT (Xca)

Arguments

Details

Each cell refers to a dissimilarity between a pair of objects. Small dissimilarities are represented by dark shades and large dissimilarities are represented by light shades. In the plot the dissimilarities are reorganized in such a way that, roughly speaking, (darkly shaded) diagonal blocks correspond to clusters in the data. Therefore, k dark blocks along its main diagonal suggest that the data contain k (as yet unfound) clusters and the size of each block represents the approximate size of the cluster.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., Hathaway, R.J., 2002. VAT: a tool for visual assessment of (cluster) tendency. Proceedings of the IEEE International Joint Conference on Neural Networks, , pp. 2225?2230.
Hathaway R.J., Bezdek J.C., 2003. Visual cluster validity for prototype generator clustering models. Pattern Recognition Letters, 24, 1563?1569.
Huband J.M., Bezdek J.C., 2008. VCV2 ? Visual Cluster Validity. In Zurada J.M., Yen G.G., Wang J. (Eds.): Lecture Notes in Computer Science, 5050, pp. 293?308. Springer-Verlag, Berlin Heidelberg.

See Also

plot.fclust, VIFCR, VCV, VCV2, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## data standardization (after removing the column Serving Size)
Mc=scale(Mc[,1:(ncol(Mc)-1)],center=TRUE,scale=TRUE)[,]
## plot of VAT
VAT(Mc)

Visual Cluster Validity

Description

Digital intensity image generated using the prototype matrix (and the membership degree matrix) to do cluster validation. The function also plots the VAT image.

Usage

 VCV (Xca, U, H, which)

Arguments

.

Details

Plot 1 (which=1): VAT. Each cell refers to a dissimilarity between a pair of objects. Small dissimilarities are represented by dark shades and large dissimilarities are represented by light shades. In the plot the dissimilarities are reorganized in such a way that, roughly speaking, (darkly shaded) diagonal blocks correspond to clusters in the data. Therefore, k dark blocks along its main diagonal suggest that the data contain k (as yet unfound) clusters and the size of each block represents the approximate size of the cluster.
Plot 2 (which=2): VCV. Each cell refers to a dissimilarity between a pair of objects computed with respect to the cluster prototypes. Small dissimilarities are represented by dark shades and large dissimilarities are represented by light shades. In the plot the dissimilarities are organized by reordering the clusters (the original first cluster is the first reordered cluster and the remaining clusters are reordered so that (new) cluster c+1 is the nearest of the remaining clusters to (newly indexed) cluster c) and the objects (in accordance with decreasing membership degrees). If k dark blocks along its main diagonal are visible, then a k-cluster structure is revealed. Note that the actual number of clusters can be revealed even when a larger number of clusters is used. This suggests that the correct value of k can sometimes be found by running the algorithm with a large value of k, and then ascertaining its correct value from the visual evidence in the VCV image.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., Hathaway, R.J., 2002. VAT: a tool for visual assessment of (cluster) tendency. Proceedings of the IEEE International Joint Conference on Neural Networks, , pp. 2225?2230.
Hathaway R.J., Bezdek J.C., 2003. Visual cluster validity for prototype generator clustering models. Pattern Recognition Letters, 24, 1563?1569.

See Also

plot.fclust, VIFCR, VAT, VCV2, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## plots of VAT and VCV
VCV(clust$Xca,clust$U,clust$H)
## plot of VCV
VCV(clust$Xca,clust$U,clust$H, 2)

(New) Visual Cluster Validity

Description

Digital intensity image generated using the membership degree matrix to do cluster validation. The function also plots the VAT image.

Usage

 VCV2 (Xca, U, which)

Arguments

.

Details

Plot 1 (which=1): VAT. Each cell refers to a dissimilarity between a pair of objects. Small dissimilarities are represented by dark shades and large dissimilarities are represented by light shades. In the plot the dissimilarities are reorganized in such a way that, roughly speaking, (darkly shaded) diagonal blocks correspond to clusters in the data. Therefore, k dark blocks along its main diagonal suggest that the data contain k (as yet unfound) clusters and the size of each block represents the approximate size of the cluster.
Plot 2 (which=2): VCV2. Each cell refers to a dissimilarity between a pair of objects computed with respect to the cluster membership degrees. Small dissimilarities are represented by dark shades and large dissimilarities are represented by light shades. In the plot the dissimilarities are reorganized by using the VAT reordering. If k dark blocks along its main diagonal are visible, then a k-cluster structure is revealed. Note that the actual number of clusters can be revealed even when a larger number of clusters is used. This suggests that the correct value of k can sometimes be found by running the algorithm with a large value of k, and then ascertaining its correct value from the visual evidence in the VCV2 image.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Bezdek J.C., Hathaway, R.J., 2002. VAT: a tool for visual assessment of (cluster) tendency. Proceedings of the IEEE International Joint Conference on Neural Networks, , pp. 2225?2230.
Huband J.M., Bezdek J.C., 2008. VCV2 ? Visual Cluster Validity. In Zurada J.M., Yen G.G., Wang J. (Eds.): Lecture Notes in Computer Science, 5050, pp. 293?308. Springer-Verlag, Berlin Heidelberg.

See Also

plot.fclust, VIFCR, VAT, VCV, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## plots of VAT and VCV2
VCV2(clust$Xca,clust$U)
## plot of VCV2
VCV2(clust$Xca,clust$U, 2)

Visual inspection of fuzzy clustering results

Description

Plots for validation of fuzzy clustering results. Three plots (selected by which) are available.

Usage

 VIFCR (fclust.obj, which)

Arguments

.

Details

Plot 1 (which=1). Histogram of the membership degrees setting breaks=seq(from=0,to=1,by=0.1). The frequencies are scaled so that the heights of the first and the latter rectangles are the same in the ideal case of crisp (non-fuzzy) memberships. The fuzzy clustering solution should be such that the heights of the first and the latter rectangles are high and those of the rectangles in the middle are low. High heights of rectangles in the middle denote the presence of ambiguous membership degrees. This is an indicator for a non-optimal clustering result.
Plot 2 (which=2). Scatter plot of the objects at the co-ordinates (u1,u2). For each object, u1 and u2 denote, respectively, the highest and the second highest membership degrees. All points lie within the triangle with vertices (0,0), (0.5,0.5) and (1,0). In the ideal case of (almost) crisp membership degrees all points are near the vertex (1,0). Points near the vertex (0.5,0.5) highlight ambiguous objects shared by two clusters. Points near the vertex (0,0) are usually outliers characterized by low membership degrees to all clusters (provided that the noise approach is considered).
Plot 3 (which=3). For each cluster, scatter plot of the of the objects at the co-ordinates (dc,uc). For each object, dc is the squared Euclidean distance between the object and the cluster prototype and uc is the membership degree of the object to the cluster. The ideal case is such that points are in the upper left area or in the lower right area. In fact, this highlights high membership degrees for small distances and low membership degrees for large distances.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Klawonn F., Chekhtman V., Janz E., 2003. Visual inspection of fuzzy clustering results. In Benitez J.M., Cordon O., Hoffmann, F., Roy R. (Eds.): Advances in Soft Computing - Engineering Design and Manufacturing, pp. 65-76. Springer, London.

See Also

plot.fclust, VAT, VCV, VCV2, unemployment

Examples

## unemployment data
data(unemployment)
## fuzzy k-means
unempFKM=FKM(unemployment,k=3,stand=1)
## all plots
VIFCR(unempFKM)
## plots 1 and 3
VIFCR(unempFKM,c(1,3))

Xie and Beni index

Description

Produces the Xie and Beni index. The optimal number of clusters k is is such that the index takes the minimum value.

Usage

 XB (Xca, U, H, m)

Arguments

Details

Xca should contain the same dataset used in the clustering algorithm, i.e., if the clustering algorithm is run using standardized data, then XB should be computed using the same standardized data.
m should be the same parameter of fuzziness used in the clustering algorithm.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Xie X.L., Beni G. (1991). A validity measure for fuzzy clustering, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 841-847.

See Also

PC, PE, MPC, SIL, SIL.F, Fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## Xie and Beni index
xb=XB(clust$Xca,clust$U,clust$H,clust$m)

Butterfly data

Description

Synthetic dataset with 2 clusters and some outliers.

Usage

data(butterfly)

Format

A matrix with 17 rows and 2 columns.

Details

The butterfly data motivate the need for the fuzzy approach to clustering.
The presence of outliers can be handled using fuzzy k-means with noise cluster. In fact, differently from fuzzy k-means, the membership degrees of the outliers are low for all the clusters.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, FKM, FKM.noise

Examples

## butterfly data
data(butterfly)
plot(butterfly,type='n')
text(butterfly[,1],butterfly[,2],labels=rownames(butterfly),cex=0.7,lwd=2)
## membership degree matrix using fuzzy k-means (rounded)
round(FKM(butterfly)$U,2)
## membership degree matrix using fuzzy k-means with noise cluster (rounded)
round(FKM.noise(butterfly,delta=3)$U,2)

Cluster membership

Description

Produces a summary of the membership degree information.

Usage

 cl.memb (U)

Arguments

Details

An object is assigned to a cluster according to the maximal membership degree. Therefore, it produces the closest hard clustering partition

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

cl.memb.H, cl.memb.t

Examples

n=20
k=3
## randomly generated membership degree matrix
U=matrix(runif(n*k,0,1), nrow=n, ncol=k)
U=U/apply(U,1,sum)
info.U=cl.memb(U)
## objects assigned to cluster 2
rownames(info.U[info.U[,1]==2,])

Cluster membership

Description

Produces a summary of the membership degree information in the hard clustering sense (objects are considered to be assigned to clusters only if the corresponding membership degree are >=0.5).

Usage

 cl.memb.H (U)

Arguments

Details

An object is assigned to a cluster according to the maximal membership degree provided that such a maximal membership degree is >=0.5, otherwise it is assumed that an object is not assigned to any cluster (denoted by cluster index = 0 in row 1).

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

cl.memb, cl.memb.t

Examples

n=20
k=3
## randomly generated membership degree matrix
U=matrix(runif(n*k,0,1), nrow=n, ncol=k)
U=U/apply(U,1,sum)
info.U=cl.memb.H(U)
## objects assigned to clusters in the hard clustering sense
rownames(info.U[info.U[,1]!=0,])

Cluster membership

Description

Produces a summary of the membership degree information according to a threshold.

Usage

 cl.memb.t (U, t)

Arguments

Details

An object is assigned to a cluster according to the maximal membership degree provided that such a maximal membership degree is >= t, otherwise it is assumed that an object is not assigned to any cluster (denoted by cluster index = 0 in row 1). The function can be useful to select the subset of objects clearly assigned to clusters (objects with maximal membership degrees >= t).

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

cl.memb, cl.memb.H

Examples

n=20
k=3
## randomly generated membership degree matrix
U=matrix(runif(n*k,0,1), nrow=n, ncol=k)
U=U/apply(U,1,sum)
## threshold t=0.6
info.U=cl.memb.t(U,0.6)
## objects clearly assigned to clusters
rownames(info.U[info.U[,1]!=0,])

Cluster size

Description

Produces the sizes of the clusters.

Usage

 cl.size (U)

Arguments

Details

An object is assigned to a cluster according to the maximal membership degree.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

cl.size.H

Examples

n=20
k=3
## randomly generated membership degree matrix
U=matrix(runif(n*k,0,1), nrow=n, ncol=k)
U=U/apply(U,1,sum)
clus.size=cl.size(U)

Cluster size

Description

Produces the sizes of the clusters in the hard clustering sense (objects are considered to be assigned to clusters only if the corresponding membership degree are >=0.5).

Usage

 cl.size.H (U)

Arguments

Details

An object is assigned to a cluster according to the maximal membership degree provided that such a maximal membership degree is >=0.5, otherwise it is assumed that an object is not assigned to any cluster.

Value

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

cl.size

Examples

n=20
k=3
## randomly generated membership degree matrix
U=matrix(runif(n*k,0,1), nrow=n, ncol=k)
U=U/apply(U,1,sum)
## cluster size in the hard clustering sense
clus.size=cl.size.H(U)

Congressional Voting Records Data

Description

1984 United Stated Congressional Voting Records for each of the U.S. House of Representatives Congressmen on the 16 key votes identified by the Congressional Quarterly Almanac.

Usage

data(houseVotes)

Format

A data.frame with 435 rows on 17 columns (16 qualitative variables and 1 classification variable).

Details

The data collect 1984 United Stated Congressional Voting Records for each of the 435 U.S. House of Representatives Congressmen on the 16 key votes identified by the Congressional Quarterly Almanac (CQA). The variable class splits the observations in democrat and republican. The qualitative variables refer to the votes on handicapped-infants, water-project-cost-sharing, adoption-of-the-budget-resolution, physician-fee-freeze, el-salvador-aid, religious-groups-in-schools, anti-satellite-test-ban, aid-to-nicaraguan-contras, mx-missile, immigration, synfuels-corporation-cutback, education-spending, superfund-right-to-sue, crime, duty-free-exports, and export-administration-act-south-africa. All these 16 variables are objects of class factor with three levels according to the CQA scheme: y refers to the types of votes ”voted for”, ”paired for” and ”announced for”; n to ”voted against”, ”paired against” and ”announced against”; yn to ”voted present”, ”voted present to avoid conflict of interest” and ”did not vote or otherwise make a position known”.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

Source

https://archive.ics.uci.edu/ml/datasets/congressional+voting+records

References

Schlimmer, J.C., 1987. Concept acquisition through representational adjustment. Doctoral dissertation, Department of Information and Computer Science, University of California, Irvine, CA.

See Also

NEFRC, NEFRC.noise

Examples

data(houseVotes)
X=houseVotes[,-1]
class=houseVotes[,1]

Plotting fuzzy clustering output

Description

Plot method for class fclust. The function creates a scatter plot visualizing the cluster structure. The objects are represented by points in the plot using observed variables or principal components.

Usage

 ## S3 method for class fclust
 ## S3 method for class 'fclust'
 plot(x, v1v2, colclus, umin, ucex, pca, ...)

Arguments

Details

In the scatter plot the objects are represented by circles (pch=16) and the prototypes by stars (pch=8) using observed variables (if pca=FALSE) or principal components (if pca=TRUE), the numbers of which are specified in v1v2. Their colors differ for every cluster according to colclus. Objects such that their maximal membership degrees are lower than umin are in black. The sizes of the circles depends on the maximal membership degrees of the corresponding objects if ucex=TRUE. Also note that principal components are extracted using standardized data.
In case of relational data, the first two components resulting from Non-metric Multidimensional Scaling performed using the package MASS are used.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

VIFCR, VAT, VCV, VCV2, Fclust, print.fclust, summary.fclust, Mc

Examples

## McDonald's data
data(Mc)
names(Mc)
## data normalization by dividing the nutrition facts by the Serving Size (column 1)
for (j in 2:(ncol(Mc)-1))
Mc[,j]=Mc[,j]/Mc[,1]
## removing the column Serving Size
Mc=Mc[,-1]
## fuzzy k-means
## (excluded the factor column Type (last column))
clust=FKM(Mc[,1:(ncol(Mc)-1)],k=6,m=1.5,stand=1)
## Scatter plot of Calories vs Cholesterol (mg)
names(Mc)
plot(clust,v1v2=c(1,5))
## Scatter plot of Calories vs Cholesterol (mg) using gray levels for the clusters
plot(clust,v1v2=c(1,5),colclus=gray.colors(6))
## Scatter plot of Calories vs Cholesterol (mg)
## coloring in black objects with maximal membership degree lower than 0.5
plot(clust,v1v2=c(1,5),umin=0.5)
## Scatter plot of Calories vs Cholesterol (mg)
## coloring in black objects with maximal membership degree lower than 0.5
## and magnifying the points according to the maximal membership degree
plot(clust,v1v2=c(1,5),umin=0.5,ucex=TRUE)
## Scatter plot using the first two principal components and
## coloring in black objects with maximal membership degree lower than 0.3
plot(clust,v1v2=1:2,umin=0.3,pca=TRUE)

Printing fuzzy clustering output

Description

Print method for class fclust.

Usage

 ## S3 method for class fclust
 ## S3 method for class 'fclust'
 print(x, ...)

Arguments

Details

The function displays the number of objects, the number of clusters, the closest hard clustering partition (objects assigned to the clusters with the highest membership degree) and the membership degree matrix (rounded).

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, summary.fclust, plot.fclust, unemployment

Examples

## unemployment data
data(unemployment)
## fuzzy k-means
unempFKM=FKM(unemployment,k=3,stand=1)
unempFKM

Summarizing fuzzy clustering output

Description

Summary method for class fclust.

Usage

 ## S3 method for class fclust
 ## S3 method for class 'fclust'
 summary(object, ...)

Arguments

Details

The function displays the number of objects, the number of clusters, the cluster sizes, the closest hard clustering partition (objects assigned to the clusters with the highest membership degree), the cluster memberships (using the closest hard clustering partition), the number of objects with unclear assignment (when the maximal membership degree is lower than 0.5), the objects with unclear assignment and the cluster sizes without unclear assignments (only if objects with unclear assignment are present), the cluster summary (for every cluster: size, minimal membership degree, maximal membership degree, average membership degree, number of objects with unclear assignment) and the Euclidean distance matrix for the cluster prototypes.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, print.fclust, plot.fclust, unemployment

Examples

## unemployment data
data(unemployment)
## fuzzy k-means
unempFKM=FKM(unemployment,k=3,stand=1)
summary(unempFKM)

Synthetic data

Description

Synthetic dataset with 2 non-spherical clusters.

Usage

data(synt.data)

Format

A matrix with 302 rows and 2 columns.

Details

Although two clusters are clearly visible, fuzzy k-means fails to discover them. The Gustafson and Kessel-like fuzzy k-means should be used for finding the known-in-advance clusters.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, FKM, FKM.gk, plot.fclust

Examples

## Not run: 
## synthetic data
data(synt.data)
plot(synt.data)
## fuzzy k-means
syntFKM=FKM(synt.data)
## Gustafson and Kessel-like fuzzy k-means
syntFKM.gk=FKM.gk(synt.data)
## plot of cluster structures from fuzzy k-means and Gustafson and Kessel-like fuzzy k-means
par(mfcol = c(2,1))
plot(syntFKM)
plot(syntFKM.gk)

## End(Not run)

Synthetic data

Description

Synthetic dataset with 2 non-spherical clusters.

Usage

data(synt.data2)

Format

A matrix with 240 rows and 2 columns.

Details

Although three clusters are clearly visible, Gustafson and Kessel - like fuzzy k-means clustering algorithm FKM.gk fails due to singularity of some covariance matrix. The Gustafson, Kessel and Babuska - like fuzzy k-means clustering algorithm FKM.gkb should be used to avoid singularity problem.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

Fclust, FKM.gk, FKM.gkb, plot.fclust

Examples

data(synt.data2)
plot(synt.data2)

## Gustafson and Kessel-like fuzzy k-means
syntFKM.gk=FKM.gk(synt.data2, k = 3, RS = 1, seed = 123)
## Gustafson, Kessel and Babuska-like fuzzy k-means
syntFKM.gkb=FKM.gkb(synt.data2, k = 3, RS = 1, seed = 123)

Unemployment data

Description

Unemployment data about some European countries in 2011.

Usage

data(unemployment)

Format

A data.frame with 32 rows and 3 columns.

Details

The source is Eurostat news-release 104/2012 - 4 July 2012. The 32 observations are European countries: BELGIUM, BULGARIA, CZECHREPUBLIC, DENMARK, GERMANY, ESTONIA, IRELAND, GREECE, SPAIN, FRANCE, ITALY, CYPRUS, LATVIA, LITHUANIA, LUXEMBOURG, HUNGARY, MALTA, NETHERLANDS, AUSTRIA, POLAND, PORTUGAL, ROMANIA, SLOVENIA, SLOVAKIA, FINLAND, SWEDEN, UNITEDKINGDOM, ICELAND, NORWAY, SWITZERLAND, CROATIA, TURKEY. The 3 variables are: the total unemployment rate, defined as the percentage of unemployed persons aged 15-74 in the economically active population (Variable 1); the youth unemployment rate, defined as the unemployment rate for young people aged between 15 and 24 (Variable 2); the long-term unemployment share, defined as the Percentage of unemployed persons who have been unemployed for 12 months or more (Variable 3). Non-spherical clusters seem to be present in the data. The Gustafson and Kessel-like fuzzy k-means should be used for finding them.

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

See Also

Fclust, FKM, FKM.gk

Examples

## unemployment data
data(unemployment)
## fuzzy k-means (only spherical clusters)
unempFKM=FKM(unemployment,k=3)
## Gustafson and Kessel-like fuzzy k-means (non-spherical clusters)
unempFKM.gk=FKM.gk(unemployment,k=3,RS=10)