OEBFDTW: Open-end/open-begin Functional Dynamic Time Warping...
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OEBFDTW

R Documentation

Open-end/open-begin Functional Dynamic Time Warping (OEB-FDTW)

Description

This function implements the OEB-FDTW.

Usage

OEBFDTW(
  x_fd,
  template_fd,
  der_x_fd,
  der_template_fd,
  alpha_vec = c(0, 0.5, 1),
  fit_c = FALSE,
  N = 100,
  M = 50,
  smin = 0.01,
  smax = 1000,
  lambda = 10^-5,
  eta = 0.5,
  iter = 3,
  delta_xs = 0,
  delta_xe = 0,
  delta_ys = 0,
  delta_ye = 0,
  der_0 = NULL,
  seq_t = NULL,
  get_fd = "no",
  n_basis_x = NULL
)

Arguments

`x_fd`	An object of class fd corresponding to the misaligned function.
`template_fd`	An object of class fd corresponding to the template function.
`der_x_fd`	An object of class fd corresponding to the first derivative of `x_fd`.
`der_template_fd`	An object of class fd corresponding to the first derivative of `template_fd`.
`alpha_vec`	Grid of values to find the optimal value of `\alpha_i`.
`fit_c`	If TRUE, the value of the objective function without the penalization evaluated at the solution is returned.
`N`	The number `N_{t}` of evenly spaced values along the template domain `\mathcal{D}_{Y}`.
`M`	The number `M_{x}` of evenly spaced values along the functional observation domain `\mathcal{D}_{X_i}`.
`smin`	The minimum values allowed for the first derivative of the warping function `h_i`. If NULL, in `FRTM_PhaseI`, it is set as 0.001 multiplied by the ratio between the size of the monitoring and template domains.
`smax`	The maximum values allowed for the first derivative of the warping function `h_i`. If NULL, in `FRTM_PhaseI`, it is set as 100 multiplied by the ratio between the size of the monitoring and template domains.
`lambda`	The smoothing parameter `\lambda_i`.
`eta`	Fraction `\eta` for updating the constraint bounds to reduce the error associated to the discretization (Deriso and Boyd, 2022).
`iter`	Number of iteration in the iterative refinement to reduce the error associated to the discretization (Deriso and Boyd, 2022).
`delta_xs`	Maximum allowed misalignment at the beginning of the process along the misaligned function domain.
`delta_xe`	Maximum allowed misalignment at the end of the process along the misaligned function domain.
`delta_ys`	Maximum allowed misalignment at the beginning of the process along the template domain.
`delta_ye`	Maximum allowed misalignment at the end of the process along the template domain.
`der_0`	The target values towards which shrinking the warping function slope. If NULL, it is equal to the ratio between the size of the domain of `x_fd` and the size of the domain of `template_fd`.
`seq_t`	Discretized sequence in the template domain `\mathcal{D}_{Y}`. If NULL, an equally spaced grid of length `N` in the template domain is used.
`get_fd`	If "x_reg", the registered function as a class fd object is returned. If "no", the registered function as a class fd object is not returned.
`n_basis_x`	Number of basis to obtain the registered function. If NULL, it is set as 0.5 the length of the optimal path.

Value

A list containing the following arguments:

mod that is a list composed by

h_fd: The estimated warping function.
x_reg: The registered function.
h_list: A list containing the discretized warping function for each iteration of the iterative refinement.
min_cost: Optimal value of the objective function.
lambda: The smoothing parameter \lambda.
alpha: Optimal value of the parameter \alpha_i.

obj Values of the objective function for each value in alpha_vec.

fit Values of the objective function without the penalization for each value in alpha_vec.

obj_opt Optimal value of the objective function.

fit_opt Optimal value of the objective function without the penalization.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

Deriso, D. and S. Boyd (2022). A general optimization framework for dynamic time warping. Optimization and Engineering, 1-22.

Examples


set.seed(1)
data <- simulate_data_FRTM(n_obs = 100)

dom <- c(0, 1)
basis_x <- fda::create.bspline.basis(c(0, 1), nbasis = 30)
x_fd <-
  fda::smooth.basis(data$grid_i[[1]] / max(data$grid_i[[1]]), data$x_err[[1]], basis_x)$fd
template_fd <-
  fda::smooth.basis(data$grid_template, data$template, basis_x)$fd
der_x_fd <- fda::deriv.fd(x_fd, 1)
der_template_fd <- fda::deriv.fd(template_fd, 1)

mod <-
  OEBFDTW(x_fd, template_fd, der_x_fd , der_template_fd, get_fd = "x_reg")

funcharts documentation built on April 3, 2025, 7:47 p.m.