AMFCC_PhaseI function

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

This function implements the design phase (Phase I) of the Adaptive Multivariate Functional Control Chart.

AMFCC_PhaseI(
  data_tra,
  data_tun = NULL,
  grid,
  q = 30,
  par_seq_list = list(10^seq(-7, 2, l = 10), c(0.5, 0.7, 0.8, 0.9, 0.99)),
  alpha_diagn = 0.05,
  alpha_mon = 0.05,
  ncores = 1
)

Arguments

• data_tra: a data frame with the training 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 grid.
  • x: concatenated vector of the observed curves.

• data_tun: a data frame with the tuning data with the same structure as data_tra. If NULL, data_tun is set to data_tra.

• grid: The vector of time points where the curves are sampled.

• q: The dimension of the set of B-spline functions.

• par_seq_list: a list with two elements. The first element is a sequence of values for the regularization parameter $\lambda$ and the second element is a sequence of percentages of the total variability to select $L$.

• alpha_diagn: Type I error probability for the diagnostic.

• alpha_mon: Type I error probability for the monitoring.

• ncores: number of cores to use for parallel computing

Returns

A list containing the following arguments:

• statistics_IC: A matrix with the values of 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.
• CL: The control limits for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.
• contributions_IC: A list where each element corresponds to a variable and is a matrix with the contributions to the Hotelling $T^2$-type statistics for each observation and parameter combination.
• p_values_combined_cont: A list where each element corresponds to a variable and is a list of two elements containing the contribution to 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.
• par_seq_list: The list of the sequences of the tuning parameters.
• q: The dimension of the set of B-spline functions.
• basis: The basis functions used for the functional data representation.
• grid: The vector of time points where the curves are sampled.
• comb_list_tot: The matrix with all the parameter combinations.
• mod_pca_list: The list of the MFPCA models for each value of lambda_s.

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