| Title: | Working with and Analyzing Functional Data of Varying Lengths |
|---|---|
| Description: | Comprehensive set of tools for analyzing and manipulating functional data with non-uniform lengths. This package addresses two common scenarios in functional data analysis: Variable Domain Data, where the observation domain differs across samples, and Partially Observed Data, where observations are incomplete over the domain of interest. 'VDPO' enhances the flexibility and applicability of functional data analysis in 'R'. See Amaro et al. (2024) <doi:10.48550/arXiv.2401.05839>. |
| Authors: | Pavel Hernandez [aut, cre], Jose Ignacio Diez [ctr], Maria Durban [ctb], Maria del Carmen Aguilera-Morillo [ctb] |
| Maintainer: | Pavel Hernandez <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.1.0 |
| Built: | 2024-10-22 07:22:27 UTC |
| Source: | CRAN |
Generates a variable domain functional regression model
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 )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 )
N |
Number of subjects. |
J |
Number of maximum observations per subject. |
nsims |
Number of simulations per the simulation study. |
Rsq |
Variance of the model. |
aligned |
If the data that will be generated is aligned or not. |
multivariate |
If TRUE, the data is generated with 2 functional variables. |
beta_index |
Index for the beta. |
use_x |
If the data is generated with x. |
use_f |
If the data is generated with f. |
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.
# 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$X_se # Noisy functional covariate# Basic usage with default parameters sim_data <- data_generator_vd() # Generate data with non-aligned domains non_aligned_data <- data_generator_vd(N = 150, J = 120, aligned = FALSE) # Generate multivariate functional data multivariate_data <- data_generator_vd(N = 200, J = 100, multivariate = TRUE) # Generate data with non-functional covariates and smooth term complex_data <- data_generator_vd( N = 100, J = 150, use_x = TRUE, use_f = TRUE ) # Generate data with a different beta function and R-squared value custom_beta_data <- data_generator_vd( N = 80, J = 80, beta_index = 2, Rsq = 0.8 ) # Access components of the generated data y <- sim_data$y # Response variable X_s <- sim_data$X_s # Noise-free functional covariate X_se <- sim_data$X_se # Noisy functional covariate
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 , where is a functional covariate of length , and is an unknown bivariate functional coefficient.
The functional basis used to model this term is the B-spline basis.
ffvd(X, grid, nbasis = c(30, 50, 30), bdeg = c(3, 3, 3))ffvd(X, grid, nbasis = c(30, 50, 30), bdeg = c(3, 3, 3))
X |
variable domain functional covariate |
grid |
observation points of the variable domain functional covariate.
If not provided, it will be |
nbasis |
number of bspline basis to be used. |
bdeg |
degree of the bspline basis used. |
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.
# Generate sample data set.seed(123) data <- data_generator_vd(beta_index = 1, use_x = FALSE, use_f = FALSE) X <- data$X_se # Specifying a custom grid custom_grid <- seq(0, 1, length.out = ncol(X)) ffvd_term_custom_grid <- ffvd(X, grid = custom_grid, nbasis = c(10, 10, 10)) # Customizing the number of basis functions ffvd_term_custom_basis <- ffvd(X, nbasis = c(10, 10, 10)) # Customizing both basis functions and degrees ffvd_term_custom <- ffvd(X, nbasis = c(10, 10, 10), bdeg = c(3, 3, 3))# Generate sample data set.seed(123) data <- data_generator_vd(beta_index = 1, use_x = FALSE, use_f = FALSE) X <- data$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))
The vd_fit function fits generalized additive functional regression models
for variable domain functional data.
vd_fit(formula, data, family = stats::gaussian(), offset = NULL)vd_fit(formula, data, family = stats::gaussian(), offset = NULL)
formula |
a formula object with at least one |
data |
a |
family |
a |
offset |
An offset vector. The default value is |
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.
# 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$fitted.values # estimated response variable # Also, a summary of the fit can be accesed using the summary function summary(res) # And a heatmap for an specific beta can be obtained using the plot function plot(res, beta_index = 1)# VARIABLE DOMAIN FUNCTIONAL DATA EXAMPLE # set seed for reproducibility set.seed(42) # generate example data data <- data_generator_vd( N = 100, J = 100, beta_index = 1, use_x = TRUE, use_f = TRUE, ) # Define a formula object that specifies the model behavior. # The formula includes a functional form of the variable 'X_se' using 'ffvd' # with a non-default number of basis functions ('nbasis' is set to c(10, 10, 10)). # Additionally, it includes a smooth function 'f' applied to 'x2' with 10 segments ('nseg = 10'), # a second-order penalty ('pord = 2'), and cubic splines ('degree = 3'). # The model also contains the linear term 'x1'. formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10)) + f(x2, nseg = 10, pord = 2, degree = 3) + x1 # We can fit the model using the data and the formula res <- vd_fit(formula = formula, data = data) # Some important parameters of the model can be accesed as follows res$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)