S3 Class for functional datasets. A class for univariate or multivariate functional dataset.

Description

S3 Class for functional datasets. A class for univariate or multivariate functional dataset.

Usage

funData(grid, data)

Arguments

Value

The function returns a S3 object of class funData, containing the grid over which the functional dataset is defined and a matrix or a list of vectors or matrices containing the functional data

See Also

gmfd_simulate

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 )
m2 <- t * ( 1 - t )^2

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
x1 <- gmfd_simulate( n, m1, rho = rho, theta = theta )
x2 <- gmfd_simulate( n, m2, rho = rho, theta = theta )

FD <- funData( t, list( x1, x2 ) )

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

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 )

Dissimilarity matrix function

Description

This function computes the dissimilarity matrix containing the distances between the curves of the functional dataset

Usage

gmfd_diss(FD, metric, p = NULL, k_trunc = NULL)

Arguments

Value

The function returns a matrix of numeric values containing the distances between the 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
x <- gmfd_simulate( n, m1, rho = rho, theta = theta )

FD <- funData( t, x )

D <- gmfd_diss( FD, metric = "L2" )

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

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 )

Simulation of a functional sample

Description

Simulate a univariate functional sample using a Karhunen Loeve expansion.

Usage

gmfd_simulate(size, mean, covariance = NULL, rho = NULL, theta = NULL)

Arguments

Value

The function returns a functional data object of type funData.

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
# with the Karhunen - Loève expansion
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
x <- gmfd_simulate( n, m1, rho = rho, theta = theta )

Two-sample hypotesis tests

Description

Performs a two sample hypotesis tests on two samples of functional data.

Usage

gmfd_test(FD1, FD2, conf.level = 0.95, stat_test, p = NULL,
  k_trunc = NULL)

Arguments

Value

The function returns a list with the following components:

statistic the value of the test statistic.

quantile the value of the quantile.

p.value the p-value for the test.

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. Statics & 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 + 0.1 * s

# Simulate the functional data
x1 <- gmfd_simulate( n, m1, rho = rho, theta = theta )
x2 <- gmfd_simulate( n, m2, rho = rho, theta = theta )
FD1 <- funData( t, x1 )
FD2 <- funData( t, x2 )
output <- gmfd_test( FD1, FD2, 0.95, "mahalanobis", p = 10^5 )

A method to plot funData objects

Description

This function performs the plot of a functional dataset stored in an object of class funData.

Usage

## S3 method for class 'funData'
plot(x, ...)

Arguments

See Also

funData

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 )
m2 <- t * ( 1 - t )^2

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
x1 <- gmfd_simulate( n, m1, rho = rho, theta = theta )
x2 <- gmfd_simulate( n, m2, rho = rho, theta = theta )

FD <- funData( t, list( x1, x2 ) )

plot(FD)