Partitions a numeric data set by using the Modified Fuzzy and Possibilistic C-Means (MFPCM) clustering algorithm (Saad & Alimi, 2009).
mfpcm(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)an object of class ‘ppclust’, which is a list consists of the following items:
a numeric matrix containing the processed data set.
a numeric matrix containing the final cluster prototypes (centers of clusters).
a numeric matrix containing the fuzzy memberships degrees of the data objects.
a numeric matrix containing the distances of objects to the final cluster prototypes.
an integer for the number of clusters.
a number for the fuzzifier.
a number for the typicality exponent.
a numeric vector containing the cluster labels found by defuzzying the fuzzy membership degrees of the objects.
a numeric vector containing the number of objects in the clusters.
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 data set x 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 ‘FCM’ 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 or fuzzifier. The default is 2.
a number greater than 1 to be used as the typicality exponent. The default is 3.
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 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.
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.
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
Modified Fuzzy and Possibilistic C Means (MFPCM) algorithm was proposed by Pal et al (1997) intented to incorporate a weight parameter to the objective function of FPCM as follows:
\(J_{MFPCM}(\mathbf{X}; \mathbf{V}, \mathbf{U}, \mathbf{T})=\sum\limits_{i=1}^n u_{ij}^m w_{ij}^m \; d^{2m}(\vec{x}_i, \vec{v}_j) + t_{ij}^\eta w_{ij}^\eta \; d^{2\eta}(\vec{x}_i, \vec{v}_j)\)
In the above ojective function, every data object is considered to has its own weight in relation to every cluster. Therefore it is expected that the weight permits to have a better classification especially in the case of noise data (Saad & Alimi, 2009). The weight is calculated with the following equation:
\(w_{ij} = exp \Bigg[- \frac{d^2(\vec{x}_i, \vec{v}_j)}{\sum\limits_{i=1}^n d^2(\vec{x}_i, \bar{v}) \frac{k}{n}} \Bigg]\)
The objective function of MFPCM is minimized by using the following update equations:
\(u_{ij} =\Bigg[\sum\limits_{j=1}^k \Big(\frac{d^2(\vec{x}_i, \vec{v}_j)}{d^2(\vec{x}_i, \vec{v}_l)}\Big)^{2m/(m-1)} \Bigg]^{-1} \;\;; 1 \leq i \leq n,\; 1 \leq l \leq k\)
\(t_{ij} =\Bigg[\sum\limits_{l=1}^n \Big(\frac{d^2(\vec{x}_i, \vec{v}_j)}{d^2(\vec{x}_i, \vec{v}_l)}\Big)^{2\eta/(\eta-1)} \Bigg]^{-1} \;\;; 1 \leq i \leq n, \; 1 \leq j \leq k\)
\(\vec{v}_{j} =\frac{\sum\limits_{i=1}^n (u_{ij}^m w_{ij}^m + t_{ij}^\eta w_{ij}^\eta) \vec{x}_i}{\sum\limits_{i=1}^n (u_{ij}^m w_{ij}^m + t_{ij}^\eta) w_{ij}^\eta} \;\;; {1\leq j\leq k}\)
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>
Saad, M. F. & Alimi, A. M. (2009). Modified fuzzy possibilistic c-means. In Proc. of the Int. Multiconference of Engineers and Computer Scientists, 1: 18-20. <ISBN:978-988-17012-2-0>
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 FCM with the initial prototypes and memberships
mfpcm.res <- mfpcm(x, centers=v, memberships=u, m=2, eta=2)
# Show the fuzzy membership degrees for the top 5 objects
head(mfpcm.res$u, 5)
# Show the possibilistic membership degrees for the top 5 objects
head(mfpcm.res$t, 5)
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