Partitions a numeric data set by using the Fuzzy Possibilistic Product Partition C-Means (FPPPCM) clustering algorithm which has been proposed by Szilagyi & Szilagyi (2014).
fpppcm(x, centers, memberships, m=2, eta=2, K=1, omega,
dmetric="sqeuclidean", pw=2, alginitv="kmpp", alginitu="imembrand",
nstart=1, iter.max=1000, con.val=1e-09,
fixcent=FALSE, fixmemb=FALSE, stand=FALSE, numseed)an object of class ‘ppclust’, which is a list consists of the following items:
a numeric matrix containing the final cluster prototypes.
a numeric matrix containing the typicality degrees of the data objects.
a numeric matrix containing the distances of objects to the final cluster prototypes.
a numeric matrix containing the processed data set.
a numeric vector containing the cluster labels found by defuzzifying the typicality degrees of the objects.
a numeric vector for the number of objects in the clusters.
an integer for the number of clusters.
a number for the used fuzziness exponent.
a number for the used typicality exponent.
a numeric vector of reference distances.
an integer vector for the number of iterations in each start of the algorithm.
an integer for the index of start that produced the minimum objective functional.
a numeric vector for the objective function values in each start of the algorithm.
a numeric vector for the execution time in each start of the algorithm.
a logical value, TRUE shows that x data set contains the standardized values of raw data.
a number for the within-cluster sum of squares for each cluster.
a number for the between-cluster sum of squares.
a number for the total within-cluster sum of squares.
a number for the total sum of squares.
a string for the name of partitioning algorithm. It is ‘PCM’ with this function.
a string for the matched function call generating this ‘ppclust’ object.
a numeric vector, data frame or matrix.
an integer specifying the number of clusters or a numeric matrix containing the initial cluster centers.
a numeric matrix containing the initial membership degrees. If missing, it is internally generated.
a number greater than 1 to be used as the fuzziness exponent. The default is 2.
a number greater than 1 to be used as the typicality exponent. The default is 2.
a number greater than 0 to be used as the weight of penalty term. The default is 1.
a numeric vector of reference distances. If missing, it is internally generated.
a string for the distance metric. The default is sqeuclidean for the squared Euclidean distances. See get.dmetrics for the alternative options.
a number for the power of Minkowski distance calculation. The default is 2 if the dmetric is minkowski.
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.
a string for the initialization of memberships degrees matrix. The default is imembrand for random sampling of initial membership degrees.
an integer for the number of starts for clustering. The default is 1.
an integer for the maximum number of iterations allowed. The default is 1000.
a number for the convergence value between the iterations. The default is 1e-09.
a logical flag to fix the initial cluster centers. 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.
a logical flag to fix the initial membership degrees. 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.
a logical flag to standardize data. Its default value is FALSE. If its value is TRUE, the data matrix x is standardized.
a seeding number to set the seed of R's random number generator.
Zeynel Cebeci, Alper Tuna Kavlak & Figen Yildiz
Fuzzy Possibilistic Product Partition C-Means (FPPPCM) clustering algorithm aimed to eliminate the effect of outliers in the other fuzzy and possibilistic clustering algorithms. The algorithm includes a probabilistic and a possibilistic term via multiplicative way instead of additive combination (Gosztolya & Szilagyi, 2015). The objective function of the algorithm as follows:
\(J_{FPPPCM}(\mathbf{X}; \mathbf{V}, \mathbf{U}, \mathbf{T})=\sum\limits_{j=1}^k \sum\limits_{i=1}^n u_{ij}^m \big[ t_{ij}^\eta \; d^2(\vec{x}_i, \vec{v}_j) + \Omega_j (1-t_{ij})^\eta \big]\)
The fuzzy membership degrees in the probabilistic part of the objective function \(J_{FPPPCM}\) is updated as follows:
\(u_{ij} = \frac{\Big[t_{ij}^\eta \; d^2(\vec{x}_i, \vec{v}_j) \; + \; \Omega_j (1-t_{ij})^\eta \Big]^{-1/(m-1)}}{\Big[ \sum\limits_{l=1}^k t_{il}^\eta \; d^2(\vec{x}_i, \vec{v}_l) \; + \; \Omega_l (1-t_{il})^\eta \Big]^{-1/(m-1)}} \;;\; 1 \leq i \leq n, \; 1 \leq j \leq k\)
The typicality degrees in the possibilistic part of the objective function \(J_{FPPPCM}\) is calculated as follows:
\(t_{ij} =\Bigg[1 + \Big(\frac{d^2(\vec{x}_i, \vec{v}_j)}{\Omega_j}\Big)^{1/(\eta -1)}\Bigg]^{-1} \;;\; 1 \leq i \leq n, \; 1 \leq j \leq k\)
\(m\) is the fuzzifier to specify the amount of fuzziness for the clustering; \(1\leq m\leq \infty\). It is usually chosen as 2.
\(\eta\) is the typicality exponent to specify the amount of typicality for the clustering; \(1\leq \eta\leq \infty\). It is usually chosen as 2.
\(\Omega\) is the possibilistic penalty term to control the variance of the clusters.
The update equation for cluster prototypes:
\(\vec{v}_j =\frac{\sum\limits_{i=1}^n u_{ij}^m \; t_{ij}^\eta \; \vec{x}_i}{\sum\limits_{i=1}^n u_{ij}^m \; t_{ij}^\eta} \;;\; 1 \leq j \leq k\)
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>
Szilagyi, L. & Szilagyi, S. M. (2014). Generalization rules for the suppressed fuzzy c-means clustering algorithm. Neurocomputing, 139:298-309. <doi:10.1016/j.neucom.2014.02.027>
Gosztolya, G. & Szilagyi, L. (2015). Application of fuzzy and possibilistic c-means clustering models in blind speaker clustering. Acta Polytechnica Hungarica, 12(7):41-56. <http://publicatio.bibl.u-szeged.hu/6151/1/2015-acta-polytechnica.pdf>
ekm,
fcm,
fcm2,
fpcm,
gg,
gk,
gkpfcm,
hcm,
pca,
pcm,
pcmr,
upfc
# Load dataset X16
data(x16)
x <- x16[,-3]
# Initialize the prototype matrix using K-means++
v <- inaparc::kmpp(x, k=2)$v
# Initialize the memberships degrees matrix
u <- inaparc::imembrand(nrow(x), k=2)$u
# Run FPPPCM
res.fpppcm <- fpppcm(x, centers=v, memberships=u, m=2, eta=2)
# Display typicality degrees
res.fpppcm$t
# Run FPPPCM for eta=3
res.fpppcm <- fpppcm(x, centers=v, memberships=u, m=2, eta=3)
# Display typicality degrees
res.fpppcm$t
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