This function builds a data frame needed to plot control charts for monitoring a monitoring a scalar quality characteristic adjusted for the effect of multivariate functional covariates based on scalar-on-function regression, as proposed in Capezza et al. (2020).

In particular, this function provides:

* the Hotelling's T^2 control chart,

* the squared prediction error (SPE) control chart,

* the scalar regression control chart.

This function calls control_charts_pca for the control charts on the multivariate functional covariates and regr_cc_sof for the scalar regression control chart.

The training data have already been used to fit the model. An optional tuning data set can be provided that is used to estimate the control chart limits. A phase II data set contains the observations to be monitored with the control charts.

control_charts_sof_pc(
  mod,
  y_test,
  mfdobj_x_test,
  mfdobj_x_tuning = NULL,
  alpha = list(T2 = 0.0125, spe = 0.0125, y = 0.025),
  limits = "standard",
  seed,
  nfold = NULL,
  ncores = 1
)

Arguments

mod

A list obtained as output from sof_pc, i.e. a fitted scalar-on-function linear regression model.

y_test

A numeric vector containing the observations of the scalar response variable in the phase II data set.

mfdobj_x_test

An object of class mfd containing the phase II data set of the functional covariates observations.

mfdobj_x_tuning

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.

alpha

A named list with three elements, named T2, spe, and codey, respectively, each containing the desired Type I error probability of the corresponding control chart (T2 corresponds to the T^2 control chart, spe corresponds to the SPE control chart, y corresponds to the scalar regression 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) for additional details. Default value is list(T2 = 0.0125, spe = 0.0125, y = 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.

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. * y_hat: the predictions of the response variable corresponding to mfdobj_x_new, * y: the same as the argument y_newgiven as input to this function, * lwr: lower limit of the 1-alphaprediction interval on the response, * pred_err: prediction error calculated as y-y_hat, * pred_err_sup: upper limit of the 1-alphaprediction interval on the prediction error, * pred_err_inf: lower limit of the 1-alpha prediction interval on the prediction error.

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[201:300, , 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[1:60]
y2 <- y[91:100]
mfdobj_x1 <- mfdobj_x[1:60]
mfdobj_x_tuning <- mfdobj_x[61:90]
mfdobj_x2 <- mfdobj_x[91:100]
mod <- sof_pc(y1, mfdobj_x1)
cclist <- control_charts_sof_pc(mod = mod,
                                y_test = y2,
                                mfdobj_x_test = mfdobj_x2,
                                mfdobj_x_tuning = mfdobj_x_tuning)
plot_control_charts(cclist)