Simulate multivariate functional data
This function provides a unified simulation structure for multivariate functional data f_1, …, f_N on one- or two-dimensional domains, based on a truncated multivariate Karhunen-Loeve representation:
f_i(t) = ∑_{m = 1}^M ρ_{i,m} ψ_m(t).
The multivariate eigenfunctions
(basis functions) ψ_m are constructed from univariate orthonormal
bases. There are two different concepts for the construction, that can be
chosen by the parameter type: A split orthonormal basis (split,
only one-dimensional domains) and weighted univariate orthonormal bases
(weighted, one- and two-dimensional domains). The scores
ρ_{i,m} in the Karhunen-Loeve representation are simulated
independently from a normal distribution with zero mean and decreasing
variance. See Details.
simMultiFunData(type, argvals, M, eFunType, ignoreDeg = NULL, eValType, N)
type |
A character string, specifying the construction method for the
multivariate eigenfunctions (either |
argvals |
A list, containing the observation points for each element of
the multivariate functional data that is to be simulated. The length of
|
M |
An integer ( |
eFunType |
A character string ( |
ignoreDeg |
A vector of integers ( |
eValType |
A character string, specifying the type of
eigenvalues/variances used for the simulation of the multivariate functions
based on the truncated Karhunen-Loeve representation. See
|
N |
An integer, specifying the number of multivariate functions to be generated. |
The parameter type defines how the eigenfunction basis for the
multivariate Karhunen-Loeve representation is constructed:
type = "split": The basis functions of an underlying 'big' orthonormal
basis are split in M parts, translated and possibly reflected. This
yields an orthonormal basis of multivariate functions with M
elements. This option is implemented only for one-dimensional domains.
type = "weighted": The multivariate eigenfunction basis consists of
weighted univariate orthonormal bases. This yields an orthonormal basis of
multivariate functions with M elements. For data on two-dimensional
domains (images), the univariate basis is constructed as a tensor product of
univariate bases in each direction (x- and y-direction).
Depending on type, the other parameters have to be specified as
follows:
The parameters M
(integer), eFunType (character string) and ignoreDeg (integer
vector or NULL) are passed to the function eFun to
generate a univariate orthonormal basis on a 'big' interval. Subsequently,
the basis functions are split and translated, such that the j-th part
of the split function is defined on the interval corresponding to
argvals[[j]]. The elements of the multivariate basis functions are
given by these split parts of the original basis functions multiplied by a
random sign σ_j in {-1,1},
j = 1, …, p.
The parameters argvals, M,
eFunType and ignoreDeg are all lists of a similar structure. They are
passed element-wise to the function eFun to generate
orthonormal basis functions for each element of the multivariate functional
data to be simulated. In case of bivariate elements (images), the
corresponding basis functions are constructed as tensor products of
orthonormal basis functions in each direction (x- and y-direction).
If the j-th element of the simulated data should be defined on a one-dimensional domain, then
argvals[[j]] is a list,
containing one vector of observation points.
M[[j]] is an
integer, specifying the number of basis functions to use for this entry.
eFunType[[j]] is a character string, specifying the type of
orthonormal basis functions to use for this entry (see eFun for
possible options).
ignoreDeg[[j]] is a vector of integers,
specifying the degrees to ignore when constructing the orthonormal basis
functions. The default value is NULL.
If the j-th element of the simulated data should be defined on a two-dimensional domain, then
argvals[[j]] is a list,
containing two vectors of observation points, one for each direction
(observation points in x-direction and in y-direction).
M[[j]]
is a vector of two integers, giving the number of basis functions for each
direction (x- and y-direction).
eFunType[[j]] is a vector of two
character strings, giving the type of orthonormal basis functions for each
direction (x- and y-direction, see eFun for possible options).
The corresponding basis functions are constructed as tensor products of
orthonormal basis functions in each direction.
ignoreDeg[[j]] is
a list, containing two integer vectors that specify the degrees to ignore
when constructing the orthonormal basis functions in each direction. The
default value is NULL.
The total number of basis functions (i.e. the
product of M[[j]] for all j) must be equal!
simData |
A |
trueFuns |
A |
trueVals |
A vector of numerics, representing the eigenvalues used for simulating the data. |
C. Happ, S. Greven (2018): Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains. Journal of the American Statistical Association, 113(522): 649-659.
oldPar <- par(no.readonly = TRUE)
# split
split <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)),
M = 5, eFunType = "Poly", eValType = "linear", N = 7)
par(mfrow = c(1,2))
plot(split$trueFuns, main = "Split: True Eigenfunctions", ylim = c(-2,2))
plot(split$simData, main = "Split: Simulated Data")
# weighted (one-dimensional domains)
weighted1D <- simMultiFunData(type = "weighted",
argvals = list(list(seq(0,1,0.01)), list(seq(-0.5,0.5,0.02))),
M = c(5,5), eFunType = c("Poly", "Fourier"), eValType = "linear", N = 7)
plot(weighted1D$trueFuns, main = "Weighted (1D): True Eigenfunctions", ylim = c(-2,2))
plot(weighted1D$simData, main = "Weighted (1D): Simulated Data")
# weighted (one- and two-dimensional domains)
weighted <- simMultiFunData(type = "weighted",
argvals = list(list(seq(0,1,0.01), seq(0,10,0.1)), list(seq(-0.5,0.5,0.01))),
M = list(c(5,4), 20), eFunType = list(c("Poly", "Fourier"), "Wiener"),
eValType = "linear", N = 7)
plot(weighted$trueFuns, main = "Weighted: True Eigenfunctions (m = 2)", obs = 2)
plot(weighted$trueFuns, main = "Weighted: True Eigenfunctions (m = 15)", obs = 15)
plot(weighted$simData, main = "Weighted: Simulated Data (1st observation)", obs = 1)
plot(weighted$simData, main = "Weighted: Simulated Data (2nd observation)", obs = 2)
par(oldPar)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.