funDist {gmfd}

Distance function

Description

This function allows you to compute the distance between two curves with the chosen metric.

Usage

funDist(FD1, FD2, metric, p = NULL, lambda = NULL, phi = NULL,
  k_trunc = NULL)

Arguments

FD1

a functional data object of type funData for the first curve

FD2

a functional data object of type funData for the second curve

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.

lambda

a vector containing the eigenvalues in descending order of the functional data from which the curves are extracted.

phi

a matrix containing the eigenfunctions of the functional data in its columns from which the curves are extracted.

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 numeric value indicating the distance between the two curves.

References

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.

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 )
}

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

# Extract two rows of the functional data
x <- funData( t, z[1, ] )
y <- funData( t, z[2, ] )

lambda <- eigen(cov(z))$values
phi <- eigen(cov(z))$vectors

d <- funDist( x, y, metric = "mahalanobis", p = 1, lambda = lambda, phi = phi )