gmfd_kmeans {gmfd}

k-means clustering algorithm

Description

This function performs a k-means clustering algorithm on an univariate or multivariate functional data using a generalization of Mahalanobis distance.

Usage

gmfd_kmeans(FD, n.cl = 2, metric, p = NULL, k_trunc = NULL)

Arguments

FD

a functional data object of type funData.

n.cl

an integer representing the number of clusters.

metric

the chosen distance to be used: "L2" for the classical L2-distance, "trunc" for the truncated Mahalanobis semi-distance, "mahalanobis" for the generalized Mahalanobis distance.

p

a positive numeric value containing the parameter of the regularizing function for the generalized Mahalanobis distance.

k_trunc

a positive numeric value representing the number of components at which the truncated mahalanobis distance must be truncated

Value

The function returns a list with the following components: cluster: a vector of integers (from 1 to n.cl) indicating the cluster to which each curve is allocated; centers: a list of d matrices (k x T) containing the centroids of the clusters

References

Martino A., Ghiglietti A., Ieva F., Paganoni A. M. (2017). A k-means procedure based on a Mahalanobis type distance for clustering multivariate functional data, MOX report 44/2017

Ghiglietti A., Ieva F., Paganoni A. M. (2017). Statistical inference for stochastic processes: Two-sample hypothesis tests, Journal of Statistical Planning and Inference, 180:49-68.

Ghiglietti A., Paganoni A. M. (2017). Exact tests for the means of gaussian stochastic processes. Statistics & Probability Letters, 131:102–107.

See Also

funDist

Examples

# Define parameters
n <- 50
P <- 100
K <- 150

# Grid of the functional dataset
t <- seq( 0, 1, length.out = P )

# Define the means and the parameters to use in the simulation
m1 <- t^2 * ( 1 - t )

rho <- rep( 0, K )
theta <- matrix( 0, K, P )
for ( k in 1:K) {
  rho[k] <- 1 / ( k + 1 )^2
  if ( k%%2 == 0 )
    theta[k, ] <- sqrt( 2 ) * sin( k * pi * t )
  else if ( k%%2 != 0 && k != 1 )
    theta[k, ] <- sqrt( 2 ) * cos( ( k - 1 ) * pi * t )
  else
    theta[k, ] <- rep( 1, P )
}

s <- 0
for (k in 4:K) {
 s <- s + sqrt( rho[k] ) * theta[k, ]
}

m2 <- m1 + s

# Simulate the functional data
x1 <- gmfd_simulate( n, m1, rho = rho, theta = theta )
x2 <- gmfd_simulate( n, m2, rho = rho, theta = theta )

# Create a single functional dataset containing the simulated datasets:
FD <- funData(t, rbind( x1, x2 ) )

output <- gmfd_kmeans( FD, n.cl = 2, metric = "mahalanobis", p = 10^6 )