funcharts (version 1.7.0)

AMFEWMA_PhaseII: Adaptive Multivariate Functional EWMA control chart - Phase II

Description

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

Usage

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

Value

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.

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

Author

C. Capezza, F. Centofanti

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.

Examples

if (FALSE) 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)