gkpfcm: Gustafson-Kessel Clustering Using PFCM
In ppclust: Probabilistic and Possibilistic Cluster Analysis

Description Usage Arguments Details Value Author(s) References See Also Examples

Partitions a numeric data set by using the Gustafson-Kessel (GK) algorithm within the PFCM (Possibilistic Fuzzy C-Means) clustering algorithm (Ojeda-Magaina et al, 2006).

gkpfcm(x, centers, memberships, m=2, eta=2, K=1, omega, a, b, 
    dmetric="sqeuclidean", pw = 2, alginitv="kmpp", 
    alginitu="imembrand", nstart=1, iter.max=1000, con.val=1e-09, 
    fixcent=FALSE, fixmemb=FALSE, stand=FALSE, numseed)

`x`	a numeric vector, data frame or matrix.
`centers`	an integer specifying the number of clusters or a numeric matrix containing the initial cluster centers.
`memberships`	a numeric matrix containing the initial membership degrees. If missing, it is internally generated.
`m`	a number greater than 1 to be used as the fuzziness exponent or fuzzifier. The default is 2.
`eta`	a number greater than 1 to be used as the typicality exponent. The default is 2.
`a`	a number for the relative importance of the fuzzy part of the objective function. The default is 1.
`b`	a number for the relative importance of the possibilistic part of the objective function. The default is 1.
`K`	a number greater than 0 to be used as the weight of penalty term. The default is 1.
`omega`	a numeric vector of reference distances. If missing, it is internally generated.
`dmetric`	a string for the distance metric. The default is sqeuclidean for the squared Euclidean distances. See `get.dmetrics` for the alternative options.
`pw`	a number for the power of Minkowski distance calculation. The default is 2 if the `dmetric` is minkowski.
`alginitv`	a string for the initialization of cluster prototypes matrix. The default is kmpp for K-means++ initialization method (Arthur & Vassilvitskii, 2007). For the list of alternative options see `get.algorithms`.
`alginitu`	a string for the initialization of memberships degrees matrix. The default is imembrand for random sampling of initial membership degrees.
`nstart`	an integer for the number of starts for clustering. The default is 1.
`iter.max`	an integer for the maximum number of iterations allowed. The default is 1000.
`con.val`	a number for the convergence value between the iterations. The default is 1e-09.
`fixcent`	a logical flag to make the initial cluster centers not changed along the different starts of the algorithm. The default is `FALSE`. If it is `TRUE`, the initial centers are not changed in the successive starts of the algorithm when the `nstart` is greater than 1.
`fixmemb`	a logical flag to make the initial membership degrees not changed along the different starts of the algorithm. The default is `FALSE`. If it is `TRUE`, the initial memberships are not changed in the successive starts of the algorithm when the `nstart` is greater than 1.
`stand`	a logical flag to standardize data. Its default value is `FALSE`. If its value is `TRUE`, the data matrix `x` is standardized.
`numseed`	a seeding number to set the seed of R's random number generator.

Gustafson-Kessel clustering within Possibilistic Fuzzy C-Means (GKPFCM) algorithm is an improvement for PFCM algorithm that consists of the modification of the distance metric for d_{ij_A}. The original PFCM uses the Euclidean distance whereas GKPFCM uses the Mahalanobis distance with GK algorithm. Babuska et al (2002) have proposed an improvement for calculating the covariance matrix \mathbf{F}_j as follows:

\mathbf{F}_j := (1 - γ) \mathbf{F}_j + γ (\mathbf{F}_0)^{1/n} \mathbf{I}

In the above equation, \mathbf{F}_j involves a weighted identity matrix. The eigenvalues λ_{ij} and the eigenvectors Φ_{ij} of the resulting matrix are calculated, and the maximum eigenvalue λ_{i,max} = max_{j}/ λ_{ij} is determined. With the obtained results, λ_{i,max} = λ_{ij}/β, \forall j, which satisfies λ_{i,max} / λ_{ij} ≥q β is calculated. Finally, \mathbf{F}_j is recomputed with the following equation:

\mathbf{F}_j = [Φ_{j,1},…, Φ_{j,n}] diag(λ_{j,1}, …, λ_{j,n}) [Φ_{j,1},…, Φ_{j,n}]^{-1} \;\; \forall j

The objective function of GKPFCM is:

J_{GK}(\mathbf{X}; \mathbf{V}, \mathbf{A}, \mathbf{U}) = ∑\limits_{i=1}^n ∑\limits_{j=1}^k u_{ij}^m d_{A_j}(\vec{x}_i, \vec{v}_j)

m is the fuzzifier to specify the amount of fuzziness for the clustering; 1≤q m≤q ∞. It is usually chosen as 2.

η is the typicality exponent to specify the amount of typicality for the clustering; 1≤q η≤q ∞. It is usually chosen as 2.

The objective function J_{GKPFCM} is minimized by using the following update equations:

u_{ij} =\Bigg[∑\limits_{j=1}^k \Big(\frac{d_{A_j}(\vec{x}_i, \vec{v}_j)}{d_{A_j}(\vec{x}_i, \vec{v}_l)}\Big)^{2/(m-1)} \Bigg]^{-1} \;\;; 1≤q i ≤q n \;,\; 1 ≤q l ≤q k

t_{ij} =\Bigg[∑\limits_{j=1}^k \Big(\frac{d_{A_j}(\vec{x}_i, \vec{v}_j))}{d_{A_j}(\vec{x}_i, \vec{v}_l))}\Big)^{2/(η-1)} \Bigg]^{-1} \;;\; 1 ≤q i ≤q n \;,\; 1 ≤q l ≤q k

\vec{v}_{j} =\frac{∑\limits_{i=1}^n (u_{ij}^m + t_{ij}^η) \vec{x}_i}{∑\limits_{i=1}^n (u_{ij}^m + t_{ij}^η)} \;\;; {1≤q j≤q k}

an object of class ‘ppclust’, which is a list consists of the following items:

`x`	a numeric matrix containing the processed data set.
`v`	a numeric matrix containing the final cluster prototypes (centers of clusters).
`u`	a numeric matrix containing the fuzzy memberships degrees of the data objects.
`d`	a numeric matrix containing the distances of objects to the final cluster prototypes.
`k`	an integer for the number of clusters.
`m`	a number for the fuzzifier.
`eta`	a number greater than 1 to be used as the typicality exponent.
`a`	a number for the relative importance of the fuzzy part of the objective function.
`b`	a number for the relative importance of the possibilistic part of the objective function.
`omega`	a numeric vector of reference distances.
`cluster`	a numeric vector containing the cluster labels found by defuzzying the fuzzy membership degrees of the objects.
`csize`	a numeric vector containing the number of objects in the clusters.
`iter`	an integer vector for the number of iterations in each start of the algorithm.
`best.start`	an integer for the index of start that produced the minimum objective functional.
`func.val`	a numeric vector for the objective function values in each start of the algorithm.
`comp.time`	a numeric vector for the execution time in each start of the algorithm.
`stand`	a logical value, `TRUE` shows that data set `x` contains the standardized values of raw data.
`wss`	a number for the within-cluster sum of squares for each cluster.
`bwss`	a number for the between-cluster sum of squares.
`tss`	a number for the total within-cluster sum of squares.
`twss`	a number for the total sum of squares.
`algorithm`	a string for the name of partitioning algorithm. It is ‘FCM’ with this function.
`call`	a string for the matched function call generating this ‘ppclust’ object.

Zeynel Cebeci & Cagatay Cebeci

Arthur, D. & Vassilvitskii, S. (2007). K-means++: The advantages of careful seeding, in Proc. of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, p. 1027-1035. <http://ilpubs.stanford.edu:8090/778/1/2006-13.pdf>

Gustafson, D. E. & Kessel, W. C. (1979). Fuzzy clustering with a fuzzy covariance matrix. In Proc. of IEEE Conf. on Decision and Control including the 17th Symposium on Adaptive Processes, San Diego. pp. 761-766. <doi:10.1109/CDC.1978.268028>

Babuska, R., van der Veen, P. J. & Kaymak, U. (2002). Improved covariance estimation for Gustafson-Kessel clustering. In Proc. of Int. Conf. on Fuzzy Systems, Hawaii, 2002, pp. 1081-1085. <https://tr.scribd.com/document/209211977/Fuzzy-and-Neural-Control>.

Ojeda-Magaina, B., Ruelas, R., Corona-Nakamura, M. A. & Andina, D. (2006). An improvement to the possibilistic fuzzy c-means clustering algorithm. In Proc. of IEEE World Automation Congress, 2006 (WAC'06). pp. 1-8. <doi:10.1109/WAC.2006.376056>

ekm, fcm, fcm2, fpcm, fpppcm, gg, gk, hcm, pca, pcm, pcmr, pfcm, upfc

## Not run: 
# Load dataset iris 
data(iris)
x <- iris[,-5]

# Initialize the prototype matrix using K-means++
v <- inaparc::kmpp(x, k=3)$v

# Initialize the memberships degrees matrix 
u <- inaparc::imembrand(nrow(x), k=3)$u

# Run FCM with the initial prototypes and memberships
gkpfcm.res <- gkpfcm(x, centers=v, memberships=u, m=2)

# Show the fuzzy membership degrees for the top 5 objects
head(gkpfcm.res$u, 5)

## End(Not run)