sim-frmfr: Sampling functional regression models with functional responses

Description

Simulation of a Functional Regression Model with Functional Response (FRMFR) comprised of an additive mix of a linear and nonlinear terms: $$Y(t) = \int_a^b X(s) \beta(s,t) ds + \Delta(X)(t) + \varepsilon(t),$$ where $X$ is a random variable in the Hilbert space of square-integrable functions in $[a, b]$, $L^2([a, b])$, $\beta$ is the bivariate kernel of the FRMFR, $\varepsilon$ is a random variable in $L^2([c, d])$, and $\Delta(X)$ is a nonlinear term.

In particular, the scenarios considered in García-Portugués et al. (2021) can be easily simulated.

Usage

r_frm_fr(n, scenario = 3, X_fdata = NULL, error_fdata = NULL,
  beta = NULL, s = seq(0, 1, l = 101), t = seq(0, 1, l = 101),
  std_error = 0.15, nonlinear = NULL, concurrent = FALSE,
  int_rule = "trapezoid", n_fpc = 50, verbose = FALSE, ...)
nl_dev(X_fdata, t = seq(0, 1, l = 101), nonlinear = NULL,
  int_rule = "trapezoid", equispaced = equispaced, verbose = FALSE)

Value

A list with the following elements:

X_fdata: functional covariates, an fdata object of length n.
Y_fdata: functional responses, 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).
nl_dev: nonlinear term, an fdata object of length n.

Arguments

n: sample size, only required when scenario is given.
scenario: an index from 1 to 3 (default) denoting one of the scenarios (S1, S2 or S3) simulated in García-Portugués et al. (2021) (see details below). If scenario = NULL, X_fdata, error_fdata, and beta have to be provided. Otherwise, X_fdata, error_fdata, and beta will be ignored.
X_fdata: sample of functional covariates $X(s)$ as fdata objects of length n, with $s$ in $[a, b]$. Defaults to NULL.
error_fdata: sample of functional errors $\varepsilon(t)$ as fdata objects of length n, with $t$ in $[c, d]$. If concurrent = TRUE, X_fdata and error_fdata must be valued in the same grid. Defaults to NULL.
beta: matrix containing the values $\beta(s, t)$, for each grid point $s$ in $[a, b]$ and $t$ in $[c, d]$. If concurrent = TRUE (see details below), a row/column vector must be introduced, valued in the same grid as error_fdata. If beta = NULL (default), scenario != NULL is required.
s, t: grid points. If X_fdata, error_fdata and beta are provided, s and t are ignored. Default to s = seq(0, 1, l = 101) and t = seq(0, 1, l = 101), respectively.
std_error: standard deviation of the random variables involved in the generation of the functional error error_fdata. Defaults to 0.15.
nonlinear: nonlinear term. Either a character string ("exp", "quadratic" or "sin") or an fdata object of length n, valued in the same grid as error_fdata. If nonlinear = NULL (default), the nonlinear part is set to zero.
concurrent: flag to consider a concurrent FLRFR (degenerate case). Defaults to FALSE.
int_rule: quadrature rule for approximating the definite unidimensional integral: trapezoidal rule (int_rule = "trapezoid") and extended Simpson rule (int_rule = "Simpson") are available. Defaults to "trapezoid".
n_fpc: number of components to be considered for the generation of functional variables. Defaults to 50.
verbose: flag to display information about the sampling procedure. Defaults to FALSE.
...: further parameters passed to r_cm2013_flmfr, r_gof2021_flmfr and
r_ik2018_flmfr, depending on the chosen scenario.
equispaced: flag to indicate if X_fdata$data is valued in an equispaced grid or not. Defaults to FALSE.

Author

Javier Álvarez-Liébana.

Details

r_frm_fr samples the above regression model, where the nonlinear term $\Delta(X)$ is computed by nl_dev. Functional covariates, errors, and $\beta$ are generated automatically from the scenarios in García-Portugués et al. (2021) when scenario != NULL (see the documentation of r_gof2021_flmfr). If scenario = NULL, covariates, errors and $\beta$ must be provided. When concurrent = TRUE, the concurrent FRMFR $$Y(t) = X(t) \beta(t) + \Delta(X)(t) + \varepsilon(t)$$ is considered.
nl_dev computes a nonlinear deviation $\Delta(X)$: $\exp(\sqrt{X(a + (t - c) ((b - a) / (d - c)))})$ (for "exp"), $(X^2 (a + (t - c) ((b - a) / (d - c))) - 1)$ ("quadratic") or $(\sin(2\pi t) - \cos(2 \pi t)) \| X \|^2$ ("sin"). Also, $\Delta(X)$ can be manually set as an fdata object of length n and valued in the same grid as error_fdata.

References

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. tools:::Rd_expr_doi("10.1111/sjos.12486")

Examples

Run this code

## Generate samples for the three scenarios

# Equispaced grids and Simpson's rule

s <- seq(0, 1, l = 101)
samp <- list()
old_par <- par(mfrow = c(3, 5))
for (i in 1:3) {
  samp[[i]] <- r_frm_fr(n = 100, scenario = i, s = s, t = s,
                        int_rule = "Simpson")
  plot(samp[[i]]$X_fdata)
  plot(samp[[i]]$error_fdata)
  plot(samp[[i]]$Y_fdata)
  plot(samp[[i]]$nl_dev)
  image(x = s, y = s, z = samp[[i]]$beta, col = viridisLite::viridis(20))
}
par(old_par)

## Linear term as a concurrent model

# The grids must be have the same number of grid points for a given
# nonlinear term and a given beta function

s <- seq(1, 2, l = 101)
t <- seq(0, 1, l = 101)
samp_c_1 <- r_frm_fr(n = 100, scenario = 3, beta = sin(t) - exp(t),
                     s = s, t = t, nonlinear = fda.usc::fdata(mdata =
                       t(matrix(rep(sin(t), 100), nrow = length(t))),
                       argvals = t),
                     concurrent = TRUE)
old_par <- par(mfrow = c(3, 2))
plot(samp_c_1$X_fdata)
plot(samp_c_1$error_fdata)
plot(samp_c_1$Y_fdata)
plot(samp_c_1$nl_dev)
plot(samp_c_1$beta)
par(old_par)

## Sample for given X_fdata, error_fdata, and beta

# Non equispaced grids with sinusoidal nonlinear term and intensity 0.5
s <- c(seq(0, 0.5, l = 50), seq(0.51, 1, l = 101))
t <- seq(2, 4, len = 151)
X_fdata <- r_ou(n = 100, t = s, alpha = 2, sigma = 4, x0 = 1:100)
error_fdata <- r_ou(n = 100, t = t, alpha = 1, sigma = 1, x0 = 1:100)
beta <- r_gof2021_flmfr(n = 100, s = s, t = t)$beta
samp_Xeps <- r_frm_fr(scenario = NULL, X_fdata = X_fdata,
                      error_fdata = error_fdata, beta = beta,
                      nonlinear = "exp", int_rule = "trapezoid")
old_par <- par(mfrow = c(3, 2))
plot(samp_Xeps$X_fdata)
plot(samp_Xeps$error_fdata)
plot(samp_Xeps$Y_fdata)
plot(samp_Xeps$nl_dev)
image(x = s, y = t, z = beta, col = viridisLite::viridis(20))
par(old_par)

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