Model fitting for variable domain functional data

Introduction

The VDPO package provides, among other tools, methods for analyzing variable domain functional data. This vignette demonstrates how to fit variable domain functional regression models using the vd_fit function, which is designed to handle various types of functional and non-functional covariates in a flexible framework.

Data Generation

library(VDPO)

We’ll start by generating sample data using the data_generator_vd function. This function creates simulated data with variable domain functional covariates and additional non-functional covariates if specified.

# Generate data with functional and non-functional covariates
data <- data_generator_vd(beta_index = 1, use_x = TRUE, use_f = TRUE)

Model Fitting

The vd_fit function is the main tool for fitting variable domain functional regression models. It supports various model specifications through a formula interface.

Basic Model with Single Functional Covariate

Let’s start with a basic model using only the functional covariate:

data <- data_generator_vd(beta_index = 1, use_x = FALSE, use_f = FALSE)
formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10))
res <- vd_fit(formula = formula, data = data)

Model with Multiple Functional Covariates

If your data contains multiple functional covariates, you can include them in the model:

data <- data_generator_vd(
  beta_index = 1,
  use_x = FALSE,
  use_f = FALSE,
  multivariate = TRUE
)
formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10)) + ffvd(Y_se, nbasis = c(10, 20, 10))
res_multi <- vd_fit(formula = formula, data = data)

Model with Functional and Non-Functional Covariates

The vd_fit function also supports including non-functional covariates, both linear and smooth terms:

data <- data_generator_vd(beta_index = 1, use_x = TRUE, use_f = TRUE)
formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10)) + f(x2, nseg = 30, pord = 2, degree = 3) + x1
res_complex <- vd_fit(formula = formula, data = data)

In this model:

ffvd(X_se, nbasis = c(10, 10, 10)) specifies the functional covariate
f(x2, nseg = 30, pord = 2, degree = 3) adds a smooth effect for x2
x1 is included as a linear term

Model Summary

You can obtain a summary of the fitted model using the summary function:

summary(res_complex)
#> 
#> Family: gaussian 
#> Link function: identity 
#> 
#> 
#> Formula:
#> NULL
#> 
#> 
#> Fixed terms: 
#>                                  x2                                     
#>   1.4678062   0.9742174  -0.1430355  -3.5265899   5.2136636 -10.5801911 
#>             
#>   6.0838833 
#> 
#> 
#> Estimated degrees of freedom:
#> Total edf     Total      <NA>      <NA>      <NA> 
#>    4.9380    4.5461    0.0001    9.4842   16.4842 
#> 
#> R-sq.(adj) =  0.958   Deviance explained = 97.5%  n = 100
#> 
#> Number of iterations: 1

Working with Non-Aligned Data

The vd_fit function can handle both aligned and non-aligned functional data. Here’s an example with non-aligned data:

data_not_aligned <- data_generator_vd(aligned = FALSE, beta_index = 1)
formula <- y ~ ffvd(X_se, nbasis = c(10, 10, 10))
res_not_aligned <- vd_fit(formula = formula, data = data_not_aligned)

Additional functionality

If you need to include an offset in your model, you can use the offset argument:

offset <- rnorm(nrow(data$X_se))
res_with_offset <- vd_fit(formula = formula, data = data, offset = offset)

Plotting the betas

A heatmap for a specific beta of the model can be obtained by using the plot function:

plot(res)

Final remarks

The vd_fit function in the VDPO package provides a flexible and powerful tool for fitting variable domain functional regression models. It supports a wide range of model specifications, including multiple functional covariates, non-functional covariates, and various distribution families. By leveraging the formula interface, users can easily specify complex models tailored to their specific analysis needs.