Data generator function for the variable domain case

Description

Generates a variable domain functional regression model

Usage

data_generator_vd(
  N = 100,
  J = 100,
  nsims = 1,
  Rsq = 0.95,
  aligned = TRUE,
  multivariate = FALSE,
  beta_index = 1,
  use_x = FALSE,
  use_f = FALSE
)

Arguments

Value

A list containing the following components:

• y: vector of length N containing the response variable.

• X_s: matrix of non-noisy functional data for the first functional covariate.

• X_se: matrix of noisy functional data for the first functional covariate

• Y_s: matrix of non-noisy functional data for the second functional covariate (if multivariate).

• Y_se: matrix of noisy functional data for the second covariate (if multivariate).

• x1: vector of length N containing the non-functional covariate (if use_x is TRUE).

• x2: vector of length N containing the observed values of the smooth term (if use_f is TRUE).

• smooth_term: vector of length N containing a smooth term (if use_f is TRUE).

• Beta: array containing the true functional coefficients.

Examples

# Basic usage with default parameters
sim_data <- data_generator_vd()

# Generate data with non-aligned domains
non_aligned_data <- data_generator_vd(N = 150, J = 120, aligned = FALSE)

# Generate multivariate functional data
multivariate_data <- data_generator_vd(N = 200, J = 100, multivariate = TRUE)

# Generate data with non-functional covariates and smooth term
complex_data <- data_generator_vd(
  N = 100,
  J = 150,
  use_x = TRUE,
  use_f = TRUE
)

# Generate data with a different beta function and R-squared value
custom_beta_data <- data_generator_vd(
  N = 80,
  J = 80,
  beta_index = 2,
  Rsq = 0.8
)

# Access components of the generated data
y <- sim_data$y # Response variable
X_s <- sim_data$X_s # Noise-free functional covariate
X_se <- sim_data$X_se # Noisy functional covariate

Defining variable domain functional data terms in vd_fit formulae

Description

Auxiliary function used to define ffvd terms within vd_fit model formulae. This term represents a functional predictor where each function is observed over a domain of varying length. The formulation is \frac{1}{T_i} \int _1^{T_i} X_i(t)\beta(t,T_i)dt, where X_i(t) is a functional covariate of length T_i, and \beta(t,T_i) is an unknown bivariate functional coefficient. The functional basis used to model this term is the B-spline basis.

Usage

ffvd(X, grid, nbasis = c(30, 50, 30), bdeg = c(3, 3, 3))

Arguments

Value

the function is interpreted in the formula of a VDPO model. list containing the following elements:

• An item named B design matrix.

• An item named X_hat smoothed functional covariate.

• An item named L_Phi and B_T 1-dimensional marginal B-spline basis used for the functional coefficient.

• An item named M matrix object indicating the observed domain of the data.

• An item named nbasis number of basis used.

Examples

# Generate sample data
set.seed(123)
data <- data_generator_vd(beta_index = 1, use_x = FALSE, use_f = FALSE)
X <- data$X_se

# Specifying a custom grid
custom_grid <- seq(0, 1, length.out = ncol(X))
ffvd_term_custom_grid <- ffvd(X, grid = custom_grid, nbasis = c(10, 10, 10))

# Customizing the number of basis functions
ffvd_term_custom_basis <- ffvd(X, nbasis = c(10, 10, 10))

# Customizing both basis functions and degrees
ffvd_term_custom <- ffvd(X, nbasis = c(10, 10, 10), bdeg = c(3, 3, 3))

Estimation of the generalized additive functional regression models for variable domain functional data

Description

The vd_fit function fits generalized additive functional regression models for variable domain functional data.

Usage

vd_fit(formula, data, family = stats::gaussian(), offset = NULL)

Arguments

Value

An object of class vd_fit. It is a list containing the following items:

• An item named fit of class sop. See sop.fit.

• An item named Beta which is the estimated functional coefficient.

• An item named theta which is the basis coefficient of Beta.

• An item named covar_theta which is the covariance matrix of theta.

• An item named M which is the number of observations points for each curve.

• An item named ffvd_evals which is the result of the evaluations of the ffvd terms in the formula.

See Also

ffvd

Examples

# VARIABLE DOMAIN FUNCTIONAL DATA EXAMPLE

# set seed for reproducibility
set.seed(42)

# generate example data
data <- data_generator_vd(
  N = 100,
  J = 100,
  beta_index = 1,
  use_x = TRUE,
  use_f = TRUE,
)

# Define a formula object that specifies the model behavior.
# The formula includes a functional form of the variable 'X_se' using 'ffvd'
# with a non-default number of basis functions ('nbasis' is set to c(10, 10, 10)).
# Additionally, it includes a smooth function 'f' applied to 'x2' with 10 segments ('nseg = 10'),
# a second-order penalty ('pord = 2'), and cubic splines ('degree = 3').
# The model also contains the linear term 'x1'.
formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10)) + f(x2, nseg = 10, pord = 2, degree = 3) + x1

# We can fit the model using the data and the formula
res <- vd_fit(formula = formula, data = data)

# Some important parameters of the model can be accesed as follows
res$Beta # variable domain functional coefficient
res$fit$fitted.values # estimated response variable

# Also, a summary of the fit can be accesed using the summary function
summary(res)

# And a heatmap for an specific beta can be obtained using the plot function
plot(res, beta_index = 1)