fanny: Fuzzy Analysis Clustering

Description

Computes a fuzzy clustering of the data into k clusters.

Usage

fanny(x, k, diss = inherits(x, "dist"), metric = "euclidean", stand = FALSE)

Arguments

data matrix or data frame, or dissimilarity matrix, depending on the value of the diss argument.

In case of a matrix or data frame, each row corresponds to an observation, and each column corresponds to a variable. All variables

integer giving the desired number of clusters. It is required that $0 < k < n/2$ where $n$ is the number of observations.

diss

logical flag: if TRUE (default for dist or dissimilarity objects), then x is assumed to be a dissimilarity matrix. If FALSE, then x is treated as a matrix of observations by variables.

metric

character string specifying the metric to be used for calculating dissimilarities between observations. The currently available options are "euclidean" and "manhattan". Euclidean distances are root sum-of-squares of differences, and manhat

stand

logical; if true, the measurements in x are standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's me

Value

an object of class "fanny" representing the clustering. See fanny.object for details.

Details

In a fuzzy clustering, each observation is ``spread out'' over the various clusters. Denote by u(i,v) the membership of observation i to cluster v. The memberships are nonnegative, and for a fixed observation i they sum to 1. The particular method fanny stems from chapter 4 of Kaufman and Rousseeuw (1990). Compared to other fuzzy clustering methods, fanny has the following features: (a) it also accepts a dissimilarity matrix; (b) it is more robust to the spherical cluster assumption; (c) it provides a novel graphical display, the silhouette plot (see plot.partition).

Fanny aims to minimize the objective function $$\sum_{v=1}^k \frac{\sum_{i=1}^n\sum_{j=1}^n u_{iv}^2 u_{jv}^2 d(i,j)}{ 2 \sum_{j=1}^n u_{jv}^2}$$ where $n$ is the number of observations, $k$ is the number of clusters and $d(i,j)$ is the dissimilarity between observations $i$ and $j$.

Examples

Run this code

## generate 25 objects, divided into two clusters, and 3 objects lying
## between those clusters.
x <- rbind(cbind(rnorm(10, 0, 0.5), rnorm(10, 0, 0.5)),
           cbind(rnorm(15, 5, 0.5), rnorm(15, 5, 0.5)),
           cbind(rnorm( 3,3.5,0.5), rnorm( 3,3.5,0.5)))
fannyx <- fanny(x, 2)
fannyx
summary(fannyx)
plot(fannyx)

data(ruspini)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))

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