T^2 and SPE control charts for multivariate functional data

This function builds a data frame needed to plot the Hotelling's T^2 and squared prediction error (SPE) control charts based on multivariate functional principal component analysis (MFPCA) performed on multivariate functional data, as Capezza et al. (2020) for the multivariate functional covariates. The training data have already been used to fit the model. An optional tuning data set can be provided to estimate the control chart limits. A phase II data set contains the observations to be monitored with the control charts.

control_charts_pca(
  pca,
  components = NULL,
  tuning_data = NULL,
  newdata,
  alpha = list(T2 = 0.025, spe = 0.025),
  limits = "standard",
  seed,
  nfold = 5,
  ncores = 1,
  tot_variance_explained = 0.9,
  single_min_variance_explained = 0
)

Arguments

pca

An object of class pca_mfd obtained by doing MFPCA on the training set of multivariate functional data.

components

A vector of integers with the components over which to project the multivariate functional data. If this is not NULL, the arguments `single_min_variance_explained` and `tot_variance_explained` are ignored. If NULL, components are selected such that the total fraction of variance explained by them is at least equal to the argument `tot_variance_explained`, where only components explaining individually a fraction of variance at least equal to the argument `single_min_variance_explained` are considered to be retained. Default is NULL.

tuning_data

An object of class mfd containing the tuning set of the multivariate functional data, used to estimate the T^2 and SPE control chart limits. If NULL, the training data, i.e. the data used to fit the MFPCA model, are also used as the tuning data set, i.e. tuning_data=pca$data. Default is NULL.

newdata

An object of class mfd containing the phase II set of the multivariate functional data to be monitored.

alpha

A named list with two elements, named T2 and spe, respectively, each containing the desired Type I error probability of the corresponding control chart. Note that at the moment you have to take into account manually the family-wise error rate and adjust the two values accordingly. See Capezza et al. (2020) and Centofanti et al. (2021) for additional details. Default value is list(T2 = 0.025, spe = 0.025).

limits

A character value. If "standard", it estimates the control limits on the tuning data set. If "cv", the function calculates the control limits only on the training data using cross-validation using calculate_cv_limits. Default is "standard".

seed

If limits=="cv", since the split in the k groups is random, you can fix a seed to ensure reproducibility. Deprecated: use set.seed() before calling the function for reproducibility.

nfold

If limits=="cv", this gives the number of groups k used for k-fold cross-validation. If it is equal to the number of observations in the training data set, then we have leave-one-out cross-validation. Otherwise, this argument is ignored.

ncores

If limits=="cv", if you want perform the analysis in the k groups in parallel, give the number of cores/threads. Otherwise, this argument is ignored.

tot_variance_explained

The minimum fraction of variance that has to be explained by the set of multivariate functional principal components retained into the MFPCA model fitted on the functional covariates. Default is 0.9.

single_min_variance_explained

The minimum fraction of variance that has to be explained by each multivariate functional principal component such that it is retained into the MFPCA model. Default is 0.

Value

A data.frame with as many rows as the number of multivariate functional observations in the phase II data set and the following columns: * one id column identifying the multivariate functional observation in the phase II data set, * one T2 column containing the Hotelling T^2 statistic calculated for all observations, * one column per each functional variable, containing its contribution to the T^2 statistic, * one spe column containing the SPE statistic calculated for all observations, * one column per each functional variable, containing its contribution to the SPE statistic, * T2_lim gives the upper control limit of the Hotelling's T^2 control chart, * one contribution_T2_*_lim column per each functional variable giving the limits of the contribution of that variable to the Hotelling's T^2 statistic, * spe_lim gives the upper control limit of the SPE control chart * one contribution_spe*_lim column per each functional variable giving the limits of the contribution of that variable to the SPE statistic.

References

Capezza C, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Control charts for monitoring ship operating conditions and CO2 emissions based on scalar-on-function regression. Applied Stochastic Models in Business and Industry, 36(3):477--500. <doi:10.1002/asmb.2507>

See also

regr_cc_fof

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:220, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates],
                         n_basis = 15,
                         lambda = 1e-2)
y <- rowMeans(air$NO2)
y1 <- y[1:100]
y_tuning <- y[101:200]
y2 <- y[201:220]
mfdobj_x1 <- mfdobj_x[1:100]
mfdobj_x_tuning <- mfdobj_x[101:200]
mfdobj_x2 <- mfdobj_x[201:220]
pca <- pca_mfd(mfdobj_x1)
cclist <- control_charts_pca(pca = pca,
                             tuning_data = mfdobj_x_tuning,
                             newdata = mfdobj_x2)
plot_control_charts(cclist)