funcharts (version 1.8.1)

RoMFCC_PhaseI_casewise: Robust Multivariate Functional Control Charts - Phase I (casewise version)

Description

It performs Phase I of the Robust Multivariate Functional Control Chart (RoMFCC), proposed by Capezza et al. (2024), applied to casewise outlier detection.

Usage

RoMFCC_PhaseI_casewise(
  mfdobj_imp,
  mfdobj_imp_tuning,
  pca_par = list(fev = 0.7),
  alpha_casewise = 0.0027,
  verbose = FALSE
)

Value

A list of the following elements that are needed in Phase II:

• T2 the Hotelling's T2 statistic values for the Phase I data set,

• SPE the SPE statistic values for the Phase I data set,

• T2_tun the Hotelling's T2 statistic values for the tuning data set,

• SPE_tun the SPE statistic values for the tuning data set,

• T2_lim the Phase II control limit of the Hotelling's T2 control chart,

• spe_lim the Phase II control limit of the SPE control chart,

• mod_pca the final RoMFPCA model fitted on the Phase I data set,

• K = K the number of selected principal components,

• T_T2_inv if a tuning data set is provided, it returns the inverse of the covariance matrix of the first K scores, needed to calculate the Hotelling's T2 statistic for the Phase II observations.

• mean_scores_tuning_rob_mean if a tuning data set is provided, it returns the robust location estimate of the scores, needed to calculate the Hotelling's T2 and SPE statistics for the Phase II observations.

Arguments

mfdobj_imp

A multivariate functional data object of class mfd, already imputed and filtered, so with no cellwise outliers. A robust multivariate principal component functional analysis is applied to the imputed dataset for dimension reduction.

mfdobj_imp_tuning

An additional functional data object of class mfd, already imputed and filtered, so with no cellwise outliers. It is used to robustly estimate the distribution of the Hotelling's T2 and SPE statistics in order to calculate control limits to prevent overfitting issues that could reduce the monitoring performance of the RoMFCC.

pca_par

A list with an argument fev, indicating a number between 0 and 1 denoting the fraction of variability that must be explained by the principal components to be selected in the RoMFPCA step. All the other arguments of this list are passed as arguments to the function rpca_mfd in the RoMFPCA step. All the arguments that are not passed take their default values. See rpca_mfd for all the arguments and their default values. Default value is list(fev = 0.7).

alpha_casewise

The overall nominal type-I error probability used to set control chart limits and to identify functional casewise outliers Default value is 0.0027.

verbose

If TRUE, it prints messages about the steps of the algorithm. Default is FALSE.

Details

Unlike the original RoMFCC implementation, this version assumes that:

• functional filter

• robust multivariate functional imputation have already been applied to the training and tuning datasets. Therefore, the input data are expected to be multivariate functional data free of cellwise outliers (casewise outliers may still be present).

References

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531--547, doi:10.1080/00401706.2024.2327346.

Examples

if (FALSE) {
library(funcharts)
set.seed(0)
dat <- simulate_data_RoMFCC(p_cellwise = 0.05,
                            p_casewise = 0.05,
                            outlier = "outlier_E",
                            M_outlier_cell = 0.03,
                            M_outlier_case = 0.01,
                            max_n_cellwise = 10)
mfdobj <- get_mfd_list(dat$X_mat_list, n_basis = 5)
mfdobj_training <- mfdobj[1:333, ]
mfdobj_tuning <- mfdobj[334:1000, ]
ff_training <- functional_filter(mfdobj = mfdobj_training)
ff_tuning <- functional_filter(mfdobj = mfdobj_tuning)
x_imp_training <- RoMFDI(mfdobj = ff_training$mfdobj)
x_imp_tuning <- RoMFDI(mfdobj = ff_tuning$mfdobj)
X_imp_training <- x_imp_training[[1]]
X_imp_tuning <- x_imp_tuning[[1]]
out_phase1_casewise <- RoMFCC_PhaseI_casewise(
  mfdobj_imp = X_imp_training,
  mfdobj_imp_tuning = X_imp_tuning
)
mfd_all_imputed <- rbind_mfd(X_imp_training, X_imp_tuning)
out_phase2_casewise <- RoMFCC_PhaseII_casewise(
  mfdobj_all_imp = mfdobj_all_imputed,
  mod_phaseI_casewise = out_phase1_casewise
)
plot_control_charts(out_phase2_casewise)
}