funcharts (version 1.7.0)

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

Description

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

Usage

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
)

Value

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.

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

Author

F. Centofanti

References

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

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)