fpcm: Fuzzy Possibilistic C-Means Clustering
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 Fuzzy and Possibilistic C-Means (FPCM) clustering algorithm (Pal et al, 1997).

fpcm(x, centers, memberships, m=2, eta=2,  
     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 3.
`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.

Fuzzy and Possibilistic C Means (FPCM) algorithm which has been proposed by Pal et al (1997) indended to combine the characteristics of FCM and PCM, and hence, was also so-called Mixed C-Means (MCM) algorithm.

The objective function of FPCM is:

J_{FPCM}(\mathbf{X}; \mathbf{V}, \mathbf{U}, \mathbf{T})=∑\limits_{i=1}^n (u_{ij}^m + t_{ij}^η) \; d^2(\vec{x}_i, \vec{v}_j)

In the above equation:

\mathbf{X} = {\vec{x}_1, \vec{x}_2,…, \vec{x}_n} \subseteq\Re^p is the data set for n objects in the p-dimensional data space \Re,

\mathbf{V} = {\vec{v}_1, \vec{v}_2, …, \vec{v}_k} \subseteq\Re^n is the protoype matrix of the clusters,

\mathbf{U} = {u_{ij}} is the matrix for a fuzzy partition of \mathbf{X},

\mathbf{T} = {t_{ij}} is the matrix for a possibilistic partition of \mathbf{X},

d^2(\vec{x}_i, \vec{v}_j) is the squared Euclidean distance between the object \vec{x}_j and cluster prototype \vec{v}_i.

d^2(\vec{x}_i , \vec{v}_j) = ||\vec{x}_i - \vec{v}_j||^2 = (\vec{x}_i - \vec{v}_j)^T (\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.

FPCM must satisfy the following constraints:

∑\limits_{j=1}^k u_{ij} = 1 \;\;;\; 1 ≤q i≤q n

∑\limits_{i=1}^n t_{ij} = 1 \;\;;\; 1 ≤q j≤q k

The objective function of FPCM is minimized by using the following update equations:

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

t_{ij} =\Bigg[∑\limits_{l=1}^n \Big(\frac{d^2(\vec{x}_i, \vec{v}_j)}{d^2(\vec{x}_i, \vec{v}_l)}\Big)^{1/(η-1)} \Bigg]^{-1} \;\;; 1≤q i ≤q n, \; 1 ≤q j ≤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 for the typicality exponent.
`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, Alper Tuna Kavlak & Figen Yildiz

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>

Pal, N.R., Pal, K., & Bezdek, J.C. (1997). A mixed c-means clustering model. In Proc. of the 6th IEEE Int. Conf. on Fuzzy Systems, 1, pp. 11-21. <doi:10.1109/FUZZY.1997.616338>

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

# 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 FPCM with the initial prototypes and memberships
fpcm.res <- fpcm(x, centers=v, memberships=u, m=2, eta=2)

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

# Show the possibilistic membership degrees for the top 5 objects
head(fpcm.res$t, 5)

  Cluster 1   Cluster 2   Cluster 3
1 0.9966245 0.001071795 0.002303750
2 0.9758559 0.007497167 0.016646895
3 0.9798258 0.006414823 0.013759338
4 0.9674288 0.010107439 0.022463755
5 0.9944697 0.001768218 0.003762131
            [,1]         [,2]         [,3]
[1,] 0.046645093 0.0002045580 0.0003973692
[2,] 0.006395542 0.0002003638 0.0004020770
[3,] 0.007035303 0.0001878231 0.0003640960
[4,] 0.004633340 0.0001973998 0.0003964988
[5,] 0.027709244 0.0002009090 0.0003863238