elem-flmfr {goffda}

Covariate, error, and kernel of a functional linear model with functional response

Description

Simulation of X, a random variable in the Hilbert space of square-integrable functions in [a, b], L^2([a, b]), and \varepsilon, a random variable in L^2([c, d]). Together with the bivariate kernel \beta, they are the necessary elements for sampling a Functional Linear Model with Functional Response (FLMFR):

Y(t) = \int_a^b X(s) \beta(s,t) ds + \varepsilon(t).

The next functions sample X and \varepsilon, and construct \beta, using different proposals in the literature:

• r_cm2013_flmfr is based on the numerical example given in Section 3 of Crambes and Mas (2013). Termed as S1 in Section 2 of García-Portugués et al. (2021).

• r_ik2018_flmfr is based on the numerical example given in Section 4 of Imaizumi and Kato (2018), but zeroing the first Functional Principal Components (FPC) coefficients of \beta (so the first FPC are not adequate for estimation). S3 in Section 2 of García-Portugués et al. (2021).

• r_gof2021_flmfr gives a numerical example in Section 2 of García-Portugués et al. (2021), denoted therein as S2.

Usage

r_cm2013_flmfr(n, s = seq(0, 1, len = 101), t = seq(0, 1, len = 101),
  std_error = 0.15, n_fpc = 50, concurrent = FALSE)

r_ik2018_flmfr(n, s = seq(0, 1, l = 101), t = seq(0, 1, l = 101),
  std_error = 1.5, parameters = c(1.75, 0.8, 2.4, 0.25), n_fpc = 50,
  concurrent = FALSE)

r_gof2021_flmfr(n, s = seq(0, 1, len = 101), t = seq(0, 1, len = 101),
  std_error = 0.35, concurrent = FALSE)

Arguments

n

number of trajectories to sample.

s, t

grid points where functional covariates and responses are valued, respectively.

std_error

standard deviation of the random variables involved in the generation of the functional error error_fdata. Defaults to 0.15.

n_fpc

number of FPC to be taken into account for the data generation. Must be greater than 4 when r_ik2018_flmfr is applied, since the first 4 FPC are null. Defaults to 50.

concurrent

flag to consider a concurrent FLMFR (degenerate case). Defaults to FALSE.

parameters

vector of parameters, only required for r_ik2018_flmfr. Defaults to
c(1.75, 0.8, 2.4, 0.25).

Details

Descriptions of the processes X and \varepsilon, and of \beta can be seen in the references.

Value

A list with the following elements:

X_fdata

functional covariates, an fdata object of length n.

error_fdata

functional errors, an fdata object of length n.

beta

either the matrix with \beta(s, t) evaluated at the argvals of X_fdata and Y_fdata (if concurrent = FALSE) or a vector with \beta(t) evaluated at the argvals of X_fdata (if concurrent = TRUE).

Author(s)

Javier Álvarez-Liébana.

References

Cardot, H. and Mas, A. (2013). Asymptotics of prediction in functional linear regression with functional outputs. Bernoulli, 19(5B):2627–2651. doi:10.3150/12-BEJ469

Imaizumi, M. and Kato, K. (2018). PCA-based estimation for functional linear regression with functional responses. Journal of Multivariate Analysis, 163:15–36. doi:10.1016/j.jmva.2017.10.001

García-Portugués, E., Álvarez-Liébana, J., Álvarez-Pérez, G. and Gonzalez-Manteiga, W. (2021). A goodness-of-fit test for the functional linear model with functional response. Scandinavian Journal of Statistics, 48(2):502–528. doi:10.1111/sjos.12486

Examples

# FLMFR based on Imaizumi and Kato (2018) adopting different Hilbert spaces
s <- seq(0, 1, l = 201)
t <- seq(2, 4, l = 301)
r_ik2018 <- r_ik2018_flmfr(n = 50, s = s, t = t, std_error = 1.5,
                           parameters = c(1.75, 0.8, 2.4, 0.25), n_fpc = 50)
plot(r_ik2018$X_fdata)
plot(r_ik2018$error_fdata)
image(x = s, y = t, z = r_ik2018$beta, col = viridisLite::viridis(20))

# FLMFR based on Cardot and Mas (2013) adopting different Hilbert spaces
r_cm2013 <- r_cm2013_flmfr(n = 50, s = s, t = t, std_error = 0.15,
                           n_fpc = 50)
plot(r_cm2013$X_fdata)
plot(r_cm2013$error_fdata)
image(x = s, y = t, z = r_cm2013$beta, col = viridisLite::viridis(20))

# FLMFR in García-Portugués et al. (2021) adopting different Hilbert spaces
r_gof2021 <- r_gof2021_flmfr(n = 50, s = s, t = t, std_error = 0.35,
                             concurrent = FALSE)
plot(r_gof2021$X_fdata)
plot(r_gof2021$error_fdata)
image(x = s, y = t, z = r_gof2021$beta, col = viridisLite::viridis(20))

# Concurrent model in García-Portugués et al. (2021)
r_gof2021 <- r_gof2021_flmfr(n = 50, s = s, t = s, std_error = 0.35,
                             concurrent = TRUE)
plot(r_gof2021$X_fdata)
plot(r_gof2021$error_fdata)
plot(r_gof2021$beta)