AMFCC_PhaseII function

Phase II of the Adaptive Multivariate Functional Control Chart (AMFCC).

This function implements the monitoring phase (Phase II) of the Adaptive Multivariate Functional Control Chart.

AMFCC_PhaseII(data = NULL, mod_Phase_I, ncores = 1)

Arguments

• data: a data frame with the testing data with the following columns:

  var: vector of the variable indexes.
  
   curve: vector of the curve indexes.
  
   timeindex: vector of the time indexes corresponding to given elements of \code{grid}.
  
   x: concatenated vector of the observed curves.
  

• mod_Phase_I: a list with the output of the Phase I.

• ncores: number of cores to use for parallel computing

Returns

A list containing the following arguments:

• ARL: The average run length (ARL) for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• ARL_cont: The average run length for the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• statistics: A matrix with the values of the Hotelling T^2-type statistics for each observation and parameter combination.
• contributions: A list where each element is a matrix with the contributions to the Hotelling T^2-type statistics for each observation and parameter combination.
• p_values_combined: A list with two elements containing the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• p_values_combined_cont: A list where each element is a list of two elements containing the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• CL: The control limits for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• CL_cont: The control limits for the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

Examples

library(funcharts)
N <- 10
l_grid <- 10
p <- 2
grid <- seq(0, 1, l = l_grid)

Xall_tra <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  correlation_type_x = c("Bessel", "Gaussian")
)
X_tra <-
  data.frame(
    x = c(Xall_tra$X_list[[1]], Xall_tra$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

Xall_II <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  shift_type_x = list("A", "B"),
  d_x = c(10, 10),
  correlation_type_x = c("Bessel", "Gaussian")
)

X_II <-
  data.frame(
    x = c(Xall_II$X_list[[1]], Xall_II$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

# AMFCC -------------------------------------------------------------------
print("AMFCC")

mod_phaseI_AMFCC <- AMFCC_PhaseI(
  data_tra = X_tra,
  data_tun =
    NULL,
  grid = grid,
  ncores = 1
)

mod_phaseII_AMFCC <- AMFCC_PhaseII(data = X_II,
mod_Phase_I = mod_phaseI_AMFCC,
ncores = 1)

plot(mod_phaseII_AMFCC)
plot(mod_phaseII_AMFCC,type='cont',ind_obs=1)

References

Centofanti, F., A. Lepore, and B. Palumbo (2025). An Adaptive Multivariate Functional Control Chart. Accepted for publication in Technometrics.

Author(s)

F. Centofanti