functional_calibration_wavelets {FunctionalCalibration}R Documentation

Functional Data Calibration with Wavelets

Description

This function performs functional calibration based on the following model:

A_i(x_m) = \displaystyle \sum_{l=1}^{L} y_{il} \alpha_l(x_m) + e_i(x_m), \quad i = 1,...,I, \quad m = 1,...,M = 2^J

where the functions \alpha_l(x) are estimated using wavelet decomposition.

In matrix notation, the model is represented as:

A = \alpha y + e

Usage

functional_calibration_wavelets(
  data,
  weights,
  wavelet = "DaubExPhase",
  method = "bayesian",
  tau = 1,
  p = NULL,
  sigma = NULL,
  MC = FALSE,
  type = "soft",
  singular = FALSE,
  x = NULL
)

Arguments

data

A matrix M x I where each column represents one sample of the aggregated function — the matrix A in the model.

weights

A matrix L x I representing the weight values associated with each sample — the matrix y in the model.

wavelet

A string indicating the wavelet family to be used in the Discrete Wavelet Transform (DWT).

method

A string specifying the shrinkage method applied to the empirical wavelet coefficients. Options are: "bayesian", "universal", "sure", "probability", or "cv".

tau

A numeric value for the \tau parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

p

A numeric value for the p parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

sigma

A numeric value for the \sigma parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

MC

A logical evaluating to TRUE or FALSE indicating if the integrals in the Bayesian shrinkage are approximated using Monte Carlo simulation.

type

A string indicating whether the thresholding should be "soft" or "hard" (applies only when the method is not "bayesian").

singular

A logical evaluating to TRUE or FALSE indicating if it adds a small constant (1e-10) to the diagonal of yy^T to stabilize the matrix inversion.

x

A numeric vector of values at which the function is evaluated. If NULL, the default is the sequence 1:nrow(data).

Value

The function returns a list containing two objects:

alpha

A matrix with the estimated functional coefficients \alpha.

Plots

A list of plot objects, each representing the corresponding function \alpha_l(x).

References

dos Santos Sousa, A. R. (2024). A wavelet-based method in aggregated functional data analysis. Monte Carlo Methods and Applications, 30(1), 19-30.

Examples

functional_calibration_wavelets(simulated_data$data, simulated_data$weights)
functional_calibration_wavelets(simulated_data$data, simulated_data$weights,
                                tau = 5, p = 0.95, sigma = 0.1, x = simulated_data$x)
functional_calibration_wavelets(simulated_data$data, simulated_data$weights,
                                method = "universal")


[Package FunctionalCalibration version 1.0.0 Index]