opt.fpcac {clustEff}

Optimal cluster selection in Functional Principal Components Analysis Clustering

Description

This function provides the optimal selection of clusters for the algorithm FPCAC, as a variant of a k-means algorithm based on the principal component rotation of data

Usage

opt.fpcac(X, k.max = 5, method = c("silhouette", "wss"),
          fd = NULL, nbasis = 5, norder = 3, nharmonics = 3,
          alpha = 0, niter = 30, Ksteps = 10, seed,
          diss = NULL, trace=FALSE)

Arguments

X

Matrix of ‘curves’ of dimension n x q.

k.max

the number of cluster used in the optimization step to select the optimal one.

method

the method used to select the optimal number of clusters, "silhouette" or "wss" (whithin sum of squares.

fd

If not NULL it overrides X and must be an object of class fd.

nbasis

an integer variable specifying the number of basis functions. The default value is 5.

norder

an integer specifying the order of b-splines, which is one higher than their degree. The default value is 3.

nharmonics

the number of harmonics or principal components to use. The default value is 3.

alpha

trimming size, that is the given proportion of observations to be discarded.

niter

the number or random restarting (larger values provide more accurate solutions.

Ksteps

the number of k-mean steps (not too many ksteps are needed).

seed

the seed used for reproducibility.

diss

the dissimilarity matrix used to compute measures "silhouette" or "wss".

trace

if TRUE, it is used to print some information across the algorithm.

Details

Silhouette is a method for validate the consistency within clusters, providing a measure of how similar an object is to its own cluster compared to other clusters. The silhouette score S belongs to the interval [-1,1]. S close to one means that the data is appropriately clustered. If S is close to negative one, datum should be clustered in its neighbouring cluster. S near zero means that the datum is on the border of two natural clusters.

The wss is obtained as the classical sum of the squared deviations from each observation and the cluster centroid, providing a measure of the variability of the observations within each cluster. Clusters with higher values exhibit greater variability of the observations within the cluster.

Value

a list containing the following items:

obj.function

the sequence of objective functions.

clusters

the matrix in which each columns identify clusters for each fixed K.

K

the sequence of K used.

K.opt

the optimal number of clusters

plot

a ggplot object to plot the curve of silhouette or whithin sum of squares.

Author(s)

Gianluca Sottile [e-mail]

References

Peter J. Rousseeuw (1987). Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis. Computational and Applied Mathematics. 20, 53-65

K. V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press.

See Also

fpcac.

Examples

set.seed(1234)
n <- 300
x <- 1:n/n

Y <- matrix(0, n, 30)

sigma2 <- 4*pmax(x-.2, 0) - 8*pmax(x-.5, 0) + 4*pmax(x-.8, 0)

mu <- sin(3*pi*x)
for(i in 1:10) Y[, i] <- mu + rnorm(length(x), 0, pmax(sigma2, 0))

mu <- cos(3*pi*x)
for(i in 11:23) Y[,i] <- mu + rnorm(length(x), 0, pmax(sigma2,0))

mu <- sin(3*pi*x)*cos(pi*x)
for(i in 24:28) Y[, i] <- mu + rnorm(length(x), 0, pmax(sigma2, 0))

mu <- 0 #sin(1/3*pi*x)*cos(2*pi*x)
for(i in 29:30) Y[, i] <- mu + rnorm(length(x), 0, pmax(sigma2, 0))

num.clust <- opt.fpcac(Y)
obj2 <- fpcac(Y, K = num.clust$K.opt, disp = FALSE)
obj2