roahd (version 1.4.3)

generate_gauss_mfdata: Generation of gaussian multivariate functional data

Description

generate_gauss_mfdata generates a dataset of multivariate functional data with a desired mean and covariance function in each dimension and a desired correlation structure among components.

Usage

generate_gauss_mfdata(
  N,
  L,
  centerline,
  correlations,
  listCov = NULL,
  listCholCov = NULL
)

Arguments

N

the number of distinct functional observations to generate.

L

the number of components of the multivariate functional data.

centerline

the centerline of the distribution, represented as a 2-dimensional data structure with L rows (one for each dimension) having the measurements along the grid as columns.

correlations

is the vector containing the \(1/2 L (L-1)\) correlation coefficients \(\rho_{ij}\) in the model generating data. They have to be provided in the following order: $$(\rho_{1,2},\ldots,\rho_{1,L},\rho_{2,3},\ldots,\rho_{2,L},\ldots, \rho_{L,L-1}),$$ that is to say, the row-wise, upper triangular part of the correlation matrix without the diagonal.

listCov

a list containing the \(L\) covariance operators (provided in form of a \(P \times P\) matrix), one for each component of the multivariate functional random variable, that have to be used in the generation of the processes \(\epsilon_1(t), \ldots, \epsilon_L(t)\). At least one argument between listCov and listCholCov must be different from NULL.

listCholCov

the Cholesky factor of the \(L\) covariance operators (in \(P \times P\) matrix form), one for each component of the multivariate functional random variable, that have to be used in the generation of the processes \(\epsilon_1(t), \ldots, \epsilon_L(t)\). At least one argument between listCov and listCholCov must be different from NULL.

Value

The function returns a list of L matrices, one for each component of the multivariate functional random variable, containing the discretized values of the generated observations (in form of \(N \times P\) matrices).

Details

In particular, the following model is considered for the generation of data:

$$X(t) = ( m_1( t ) + \epsilon_1( t ), \ldots, m_L(t) + \epsilon_L(t)), \quad t \in I = [a, b]$$

where \(L\) is the number of components of the multivariate functional random variable, \(m_i(t)\) is the \(i-\)th component of the center and \(\epsilon_i(t)\) is a centered gaussian process with covariance function \(C_i\). That is to say:

$$Cov( \epsilon_{i}(s), \epsilon_{i}(t) ) = C( s, t ), \quad \forall i = 1, \ldots, L, \quad \forall s, t \in I$$

A correlation structure among \(\epsilon_1(t),\ldots,\epsilon_L(t)\) is allowed in the following way:

$$ Cor( \epsilon_i(t), \epsilon_j(t)) = \rho_{i,j}, \quad \forall i \neq j, \quad \forall t \in I.$$

All the functions are supposed to be observed on an evenly-spaced, one- dimensional grid of P points: \([ a = t_0, t_1, \ldots, t_{P-1} = b] \subset I \).

See Also

exp_cov_function, mfData, generate_gauss_fdata

Examples

# NOT RUN {
N = 30
P = 1e2
L = 3

time_grid = seq( 0, 1, length.out = P )

C1 = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 )
C2 = exp_cov_function( time_grid, alpha = 0.2, beta = 0.5 )
C3 = exp_cov_function( time_grid, alpha = 0.3, beta = 1 )


centerline = matrix( c( sin( 2 * pi * time_grid ),
                        sqrt( time_grid ),
                        10 * ( time_grid - 0.5 ) * time_grid ),
                     nrow = 3, byrow = TRUE )

generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                       listCov = list( C1, C2, C3 ) )

CholC1 = chol( C1 )
CholC2 = chol( C2 )
CholC3 = chol( C3 )

generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                      listCholCov = list( CholC1, CholC2, CholC3 ) )

# }