`R/04_phaseII.R`

`control_charts_sof_pc.Rd`

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 T2 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
)
```

- 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 T2 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 T2 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.

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 T2 statistic calculated
for all observations,

* one column per each functional variable, containing its contribution to the T2 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 T2 control chart,

* one `contribution_T2_*_lim`

column per each
functional variable giving the
limits of the contribution of that variable to the
Hotelling's T2 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_new`

given as input to this function,

* `lwr`

: lower limit of the `1-alpha`

prediction interval on the response,

* `pred_err`

: prediction error calculated as `y-y_hat`

,

* `pred_err_sup`

: upper limit of the `1-alpha`

prediction interval on the prediction error,

* `pred_err_inf`

: lower limit of the `1-alpha`

prediction interval on the prediction error.

```
if (FALSE) {
#' 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)
}
```