In this vignette we show how to use the `funcharts`

package to apply the methods proposed in Centofanti et al. (2020) to build the functional regression control chart, when we have a functional response variable and multivariate functional covariates. Let us show how the `funcharts`

package works through an example with the dataset `air`

, which has been included from the R package `FRegSigCom`

and is used in the paper of Qi and Luo (2019). The authors propose a function-on-function regression model of the `NO2`

functional variable on all the other functional variables available in the dataset.

NOTE: since the objective of this vignette is only to illustrate how the package works, in the following we will use only 15 basis functions and a fixed smoothing parameter to reduce the computational time.

First of all, starting from the discrete data, let us build the multivariate functional data objects of class `mfd`

, see `vignette("mfd")`

.

```
library(funcharts)
data("air")
<- names(air)[names(air) != "NO2"]
fun_covariates <- get_mfd_list(air,
mfdobj grid = 1:24,
n_basis = 15,
lambda = 1e-2)
<- mfdobj[, "NO2"]
mfdobj_y <- mfdobj[, fun_covariates] mfdobj_x
```

In order to perform the statistical process monitoring analysis, we divide the data set into a phase I and a phase II dataset.

```
<- 1:300
rows1 <- 301:355
rows2 <- mfdobj_x[rows1]
mfdobj_x1 <- mfdobj_x[rows2]
mfdobj_x2 <- mfdobj_y[rows1]
mfdobj_y1 <- mfdobj_y[rows2] mfdobj_y2
```

We can build a function-on-function linear regression model using the function `fof_pc`

, which is based on the multivariate functional principal component analysis (MFPCA) on the multivariate functional covariates and the functional response.

The components to retain in the MFPCA models are selected according to the total variance explained criterion, which can be set in the arguments `tot_variance_explained_x`

, `tot_variance_explained_y`

and `tot_variance_explained_res`

(for the residuals). One can also set the type_residuals in order to choose between standard residuals and studentized residuals (see Centofanti et al. (2020)).

`<- fof_pc(mfdobj_y = mfdobj_y1, mfdobj_x = mfdobj_x1) mod_fof `

As a result you get a list with the original data used for model estimation, the result of applying `pca_mfd`

on the multivariate functional covariates, the functional response, and the residuals, the estimated regression model and additional information. It is possible to plot the estimated functional regression coefficients, which is a bivariate functional data object of class `bifd`

using the function `plot_bifd`

:

`plot_bifd(mod_fof$beta_fd)`

The function `regr_cc_fof`

provides a data frame with all the information required to plot the desired functional regression control chart (see Centofanti et al. (2020)). Among the arguments, you can pass a tuning data set with the arguments `mfdobj_y_tuning`

and `mfdobj_x_tuning`

, which is not used for model estimation/training, but is used only to estimate control chart limits. If this data set is not provided, control chart limits are calculated on the basis of the training data. The arguments `mfdobj_y_new`

and `mfdobj_x_new`

contain the phase II data set of observations of the functional response and multivariate functional covariates that are to be monitored with the control charts. The function `plot_control_charts`

returns a ggplot with the control charts.

```
<- regr_cc_fof(object = mod_fof,
frcc_df mfdobj_y_new = mfdobj_y2,
mfdobj_x_new = mfdobj_x2)
plot_control_charts(frcc_df)
```

We can also plot the new functions to be monitored, against the reference training data set, by first using `predict_fof_pc`

, which produces the prediction of new observations of the functional response given the new observations of the functional covariates, as well as the corresponding prediction error:

```
<- predict_fof_pc(object = mod_fof,
y_hat mfdobj_y_new = mfdobj_y2,
mfdobj_x_new = mfdobj_x2)
```

and then using `plot_mon`

to plot a given observation against the reference data set of prediction errors/residuals used to estimate the control chart limits. Here for example we consider the observation 54

```
plot_mon(cclist = frcc_df,
fd_train = mod_fof$residuals,
fd_test = y_hat$pred_error[54])
```

As for the scalar-on-function case shown in Capezza et al. (2020), we also provide the real-time version of the functional regression control chart. Given the domain interval \((a, b)\) of the functional data, for each current domain point \(k\) to be monitored, it filters the available functional data in the interval \((a,k)\). The function `get_mfd_list_real_time`

gives a list of functional data objects each evolving up to the intermediate domain point \(k\).

```
<- get_mfd_list_real_time(data_list = air,
mfd_list grid = 1:24,
n_basis = 15,
lambda = 1e-2,
k_seq = seq(0.5, 1, length.out = 7))
<- lapply(mfd_list, function(x) x[rows1, fun_covariates])
mfd_list_x1 <- lapply(mfd_list, function(x) x[rows2, fun_covariates])
mfd_list_x2 <- lapply(mfd_list, function(x) x[rows1, "NO2"])
mfd_list_y1 <- lapply(mfd_list, function(x) x[rows2, "NO2"]) mfd_list_y2
```

Then, the function `fof_pc_real_time`

applies the function `fof_pc`

to each element in `mfd_list_x1`

and `mfd_list_x2`

.

```
<- fof_pc_real_time(
mod_fof_pc_real_time_list mfdobj_y_list = mfd_list_y1,
mfdobj_x_list = mfd_list_x1)
```

Then, we can use `control_charts_fof_pc_real_time`

to apply `control_charts_fof_pc`

to each element in `mod_fof_pc_real_time_list`

and produce control charts for the phase II data `mfd_list_y2`

and `mfd_list_x2`

.

```
<- regr_cc_fof_real_time(
cc_list_real_time mod_list = mod_fof_pc_real_time_list,
mfdobj_y_new_list = mfd_list_y2,
mfdobj_x_new_list = mfd_list_x2
)
```

Finally, we can plot the real-time control charts for a single observations, giving for each \(k\) the monitoring statistics calculated on the data observed in \((a, k)\). Here follows an example showing the real time control charts for a single phase II observation (id number 54).

`plot_control_charts_real_time(cc_list_real_time, id_num = 54)`

- 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. https://doi.org/10.1002/asmb.2507 - Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Functional Regression Control Chart.
*Technometrics*. https://doi.org/10.1080/00401706.2020.1753581 - Qi X, Luo R. (2019). Nonlinear function-on-function additive model with multiple predictor curves.
*Statistica Sinica*, 29:719–739. https://doi.org/10.5705/ss.202017.0249