OEBFDTW function

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

This function implements the OEB-FDTW.

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.

Returns

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.

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")

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.

Author(s)

F. Centofanti