# k-means clustering algorithm This function performs a k-means clustering algorithm on an univariate or multivariate functional data using a generalization of Mahalanobis distance. ```r 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 ## Returns 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 ## Examples ```r # 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 ) ``` ## 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`

gmfd_kmeans function