AMFEWMA_PhaseII function

Adaptive Multivariate Functional EWMA control chart - Phase II

This function performs Phase II of the Adaptive Multivariate Functional EWMA (AMFEWMA) control chart proposed by Capezza et al. (2024)

AMFEWMA_PhaseII(mfdobj_2, mod_1, n_seq_2 = 1, l_seq_2 = 2000)

Arguments

• mfdobj_2: An object of class mfd containing the Phase II multivariate functional data set, to be monitored with the AMFEWMA control chart.

• mod_1: The output of the Phase I achieved through the AMFEWMA_PhaseI function.

• n_seq_2: If it is 1, the Phase II monitoring statistic is calculated on the data sequence. If it is an integer number larger than 1, a number n_seq_2 of bootstrap sequences are sampled with replacement from mfdobj_2

  to allow uncertainty quantification on the estimation of the run length. Default value is 1.

• l_seq_2: If n_seq_2 is larger than 1, this parameter sets the length of each bootstrap sequence to be generated. Default value is 2000 (which is ignored if the default value

Returns

A list with the following elements.

• ARL_2: the average run length estimated over the bootstrap sequences. If n_seq_2 is 1, it is simply the run length observed over the Phase II sequence, i.e., the number of observations up to the first alarm,
• RL: the run length observed over the Phase II sequence, i.e., the number of observations up to the first alarm,
• V2: a list with length n_seq_2, containing the AMFEWMA monitoring statistic in Equation (8) of Capezza et al. (2024), calculated in each bootstrap sequence, until the first alarm.
• cc: a data frame with the information needed to plot the AMFEWMA control chart in Phase II, with the following columns. id contains the id of each multivariate functional observation, amfewma_monitoring_statistic contains the AMFEWMA monitoring statistic values calculated on the Phase II sequence, amfewma_monitoring_statistic_lim is the upper control limit.

Examples

## Not run:
set.seed(0)
library(funcharts)
dat_I <- simulate_mfd(nobs = 1000,
                      correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                      sd_x = c(0.3, 0.3, 0.3))
dat_tun <- simulate_mfd(nobs = 1000,
                        correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                        sd_x = c(0.3, 0.3, 0.3))
dat_II <- simulate_mfd(nobs = 200,
                       correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                       shift_type_x = c("C", "C", "C"),
                       d_x = c(2, 2, 2),
                       sd_x = c(0.3, 0.3, 0.3))
mfdobj_I <- get_mfd_list(dat_I$X_list)
mfdobj_tun <- get_mfd_list(dat_tun$X_list)
mfdobj_II <- get_mfd_list(dat_II$X_list)

p <- plot_mfd(mfdobj_I[1:100])
lines_mfd(p, mfdobj_II, col = "red")

mod <- AMFEWMA_PhaseI(mfdobj = mfdobj_I, mfdobj_tuning = mfdobj_tun)
print(mod$lambda)
print(mod$k)
cc <- AMFEWMA_PhaseII(mfdobj_2 = rbind_mfd(mfdobj_I[1:100], mfdobj_II),
                      mod_1 = mod)
plot_control_charts(cc$cc, nobsI = 100)
## End(Not run)

References

Capezza, C., Capizzi, G., Centofanti, F., Lepore, A., Palumbo, B. (2025) An Adaptive Multivariate Functional EWMA Control Chart. Journal of Quality Technology, 57(1):1--15, doi:https://doi.org/10.1080/00224065.2024.2383674.

Author(s)

C. Capezza, F. Centofanti