FRTM_PhaseI function

Phase I of the FRTM method.

This function implements the design phase (Phase I) of FRTM method.

FRTM_PhaseI(
  data_tra,
  data_tun = NULL,
  alpha = 0.05,
  n_basis_xall = 30,
  control.FDTW = list(),
  control.mFPCA = list(),
  control.rtr = list(),
  ncores = 1,
  print = TRUE
)

Arguments

• data_tra: A list containing the following arguments: x_err a list containing the discrete observations for each curve of the training set; grid_i a list of vector of time points where the curves of the training set are sampled.
• data_tun: A list containing the following arguments: grid_i a list containing the discrete observations for each curve of the tuning set; grid_i a list of vector of time points where the curves of the tuning set are sampled. If NULL, the tuning set is not used.
• alpha: Overall type I error probability to obtain the control chart limits.
• n_basis_xall: Number of basis to obtain the functional observation via the spline smoothing approach based on cubic B-splines and a roughness penalty on the second derivative.
• control.FDTW: A list of control parameters for the open-end/open-begin functional dynamic time warping to replace defaults returned by par.FDTW. Values not set assume default values.
• control.mFPCA: A list of control parameters for the mixed functional principal component analysis to replace defaults returned by par.mFPCA. Values not set assume default values.
• control.rtr: A list of control parameters for the real-time registration step to replace defaults returned by par.rtr. Values not set assume default values.
• ncores: If ncores>1, then parallel computing is used, with ncores cores. Default is 1.
• print: If TRUE, some information are printed. Default is TRUE.

Returns

A list containing the following arguments:

T2_fd List of $T^{2}$ functions for each observation in the tuning set.

SPE_fd List of SPE functions for each observation in the tuning set.

CL_T2 Control limit of the Hotelling's $T^{2}$ control chart.

CL_SPE Control limit of the SPE control chart.

template_fd Template function used in the registration.

der_template_fd First derivative of the template function.

u_fd Upper extreme of the band constraint.

l_fd Lower extreme of the band constraint.

x_list_tun List, for each observation in the tuning set, of partial registered functions.

h_list_tun List, for each observation in the tuning set, of partial warping functions.

x_list List, for each observation in the training set, of partial registered functions.

h_list List, for each observation in the training set, of partial warping functions.

x_err A list containing the discrete observations for each curve of the training set.

grid_i A list of vector of time points where the curves of the training set are sampled.

x_list_smooth Smooth curves from the training set.

lambda Lambda identified through the average curve distance to obtain the OEB-FDTW solution.

par_reg Additional parameters to be used in the monitoring phase (Phase II).

Examples

library(funcharts)
data <- simulate_data_FRTM(n_obs = 20)

data_oc <-
  simulate_data_FRTM(
    n_obs = 2,
    scenario = "1",
    shift = "OC_h",
    severity = 0.5
  )

lambda <- 10 ^ -5
max_x <- max(unlist(data$grid_i))
seq_t_tot <- seq(0, 1, length.out = 30)[-1]
seq_x <- seq(0.1, max_x, length.out = 10)

## Not run:

  mod_phaseI_FRTM <- FRTM_PhaseI(
    data_tra =  data,
    control.FDTW = list(
      M = 30,
      N = 30,
      lambda = lambda,
      seq_t = seq_t_tot,
      iter_tem = 1,
      iter = 1
    ),
    control.rtr = list(seq_x = seq_x)
  )
  mod_phaseII_FRTM <- FRTM_PhaseII(data_oc = data_oc , mod_phaseI = mod_phaseI_FRTM)

  plot(mod_phaseI_FRTM)
  plot(mod_phaseII_FRTM)
## End(Not run)

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.

See Also

FRTM_PhaseI

Author(s)

F. Centofanti