funcharts (version 1.8.0)

AMFEWMA_PhaseI: Adaptive Multivariate Functional EWMA control chart - Phase I

Description

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

Usage

AMFEWMA_PhaseI(
  mfdobj,
  mfdobj_tuning,
  lambda = NULL,
  k = NULL,
  ARL0 = 200,
  bootstrap_pars = list(n_seq = 200, l_seq = 2000),
  optimization_pars = list(lambda_grid = c(0.1, 0.2, 0.3, 0.5, 1), k_grid = c(1, 2, 3,
    4), epsilon = 0.1, sd_small = 0.25, sd_big = 2),
  discrete_grid_length = 25,
  score_function = "huber",
  fev = 0.9,
  n_skip = 100
)

Value

A list with the following elements. lambda is the selected lambda parameter. k is the selected k parameter. mod_1 contains the estimated Phase I model. It is a list with the following elements.

• mfdobj the mfdobj object passed as input to this function,

• mfdobj_tuning the mfdobj_tuning object passed as input to this function,

• inv_sigmaY_reg: the matrix containing the discretized version of the function K^*(s,t) defined in Equation (9) of Capezza et al. (2024),

• mean_mfdobj: the estimated mean function,

• h: the calculated upper control limit of the AMFEWMA control chart,

• ARL0: the estimated in-control ARL, which should be close to the nominal value passed as input to this function,

• lambda: the lambda parameter selected by the optimization procedure described in Section 2.4 of Capezza et al. (2024).

• k: The function C_j(t)=k sigma_j(t) appearing in the score functions (7) and (8) of Capezza et al. (2024).

• grid_points: the grid containing the points over which the functional data are discretized before computing the AMFEWMA monitoring statistic and estimating all the model parameters.

• V2_mat: the n_seqXl_seq matrix containing, in each column, the AMFEWMA monitoring statistic values of each bootstrap sequence. This matrix is used to set the control chart limit h to ensure that the desired average run length is achieved.

• n_skip: the n_skip input parameter passed to this function,

• huber: if the input parameter score_function is "huber", this is TRUE, else is FALSE,

• vectors: the discretized eigenfunctions psi_l(t) of the covariance function, appearing in Equation (9) of Capezza et al. (2024).

• values: the eigenvalues rho_l of the covariance function, appearing in Equation (9) of Capezza et al. (2024).

Arguments

mfdobj

An object of class mfd containing the Phase I multivariate functional data set, to be used to train the multivariate functional principal component analysis model.

mfdobj_tuning

An object of class mfd containing the Phase I multivariate functional data set, to be used as tuning data set to estimate the AMFEWMA control chart limit.

lambda

lambda parameter to be used in the score function. See Equation (7) or (8) of Capezza et al. (2024). If it is provided, it must be a number between zero and one. If NULL, it is chosen through the selected according to the optimization procedure presented in Section 2.4 of Capezza et al. (2024). In this case, it is chosen among the values of optimization_pars$lambda_grid. Default value is NULL.

k

k parameter to be used in the score function. See Equation (7) or (8) of Capezza et al. (2024). If it is provided, it must be a number greater than zero. If NULL, it is chosen through the selected according to the optimization procedure presented in Section 2.4 of Capezza et al. (2024). In this case, it is chosen among the values of optimization_pars$k_grid. Default value is NULL.

ARL0

The nominal in-control average run length. Default value is 200.

bootstrap_pars

Parameters of the bootstrap procedure described in Section 2.4 of Capezza et al. (2024) for the estimation of the control chart limit. It must be a list with two arguments. n_seq is the number of bootstrap sequences to be generated. l_seq is the length of each bootstrap sequence, i.e., the number of observations to be sampled with replacement from the tuning set. Default value is list(n_seq = 200, l_seq = 2000).

optimization_pars

Parameters to be used in the optimization procedure described in Section 2.4 of Capezza et al. (2024) for the selection of the parameters lambda and k. It must be a list of the following parameters. lambda_grid contains the possible values of the parameter lambda. k_grid contains the possible values of the parameter k. epsilon is the parameter used in Equation (10) of Capezza et al. (2024). When performing the parameter optimization, first the parameters lambda and k are selected to minimize the ARL with respect to a large shift, then the same parameters are chosen to minimize the ARL with respect to a small shift, given that the resulting ARL with respect to the previous large shift does not increase, in percentage, more than epsilon*100. Default value is 0.1. sd_small is a positive constant that multiplies the standard deviation function to define the small shift delta_1 in Section 2.4 of Capezza et al. (2024). In fact, the small shift is defined as delta_1(t) = mu_0(t) + sd_small * sigma(t), where mu_0(t) is the estimated in-control mean function and sigma(t) is the estimated standard deviation function. Default value is 0.25. sd_big is a positive constant that multiplies the standard deviation function to define the large shift delta_2 in Section 2.4 of Capezza et al. (2024). In fact, the large shift is defined as delta_2(t) = mu_0(t) + sd_large * sigma(t), where mu_0(t) is the estimated in-control mean function and sigma(t) is the estimated standard deviation function. Default value is 2.

discrete_grid_length

The number of equally spaced argument values at which the mfd objects are discretized. Default value is 25.

score_function

Score function to be used in Equation (7) or (8) of Capezza et al. (2024), to calculate the weighting parameter of the EWMA statistic for each observation of the sequence. Two values are possible. If "huber", it uses the score function (7) inspired by the Huber's function. If "tukey", it uses the score function (8) inspired by the Tukey's bisquare function.

fev

Number between 0 and 1 denoting the fraction of variability that must be explained by the principal components to be selected after applying multivariate functional principal component analysis on mfdobj. Default is 0.9.

n_skip

The upper control limit of the AMFEWMA control chart is set to achieve a desired in-control ARL, evaluated after the monitoring statistic has reached steady state. A monitoring statistic is in a steady state if the process has been in control long enough for the effect of the starting value to become negligible (Lucas and Saccucci, 1990). In this regard, the first n_skip observations are excluded from the calculation of the run length. Default value is 100.

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.

Lucas, J. M., Saccucci, M. S. (1990) Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12.

Examples

# \donttest{
set.seed(0)
library(funcharts)
dat_I <- simulate_mfd(nobs = 200,
                      correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                      sd_x = c(0.3, 0.3, 0.3))
dat_tun <- simulate_mfd(nobs = 200,
                        correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                        sd_x = c(0.3, 0.3, 0.3))
dat_II <- simulate_mfd(nobs = 20,
                       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$X_list, lambda = 1e-2)
mfdobj_tun <- get_mfd_list(dat_tun$X_list, lambda = 1e-2)
mfdobj_II <- get_mfd_list(dat_II$X_list, lambda = 1e-2)

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


mod <- AMFEWMA_PhaseI(mfdobj = mfdobj_I,
                      mfdobj_tuning = mfdobj_tun,
                      lambda = 0.1,
                      k = c(1, 2))

cc <- AMFEWMA_PhaseII(mfdobj_2 = rbind_mfd(mfdobj_I[1:100], mfdobj_II),
                      mod_1 = mod)
plot_control_charts(cc$cc, nobsI = 100)
# }