Function-on-function regression based on principal components

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. (2021)).

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

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)

Plotting the functional regression control chart

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. (2021)). 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.

frcc_df <- regr_cc_fof(object = mod_fof,
                       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:

y_hat <- predict_fof_pc(object = mod_fof,
                        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])

Real-time functional regression control chart

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 \(k\).

mfd_list <- get_mfd_list_real_time(data_list = air,
                                   grid = 1:24,
                                   n_basis = 15,
                                   lambda = 1e-2,
                                   k_seq = seq(0.5, 1, length.out = 7))

mfd_list_x1 <- lapply(mfd_list, function(x) x[rows1, fun_covariates])
mfd_list_x2 <- lapply(mfd_list, function(x) x[rows2, fun_covariates])
mfd_list_y1 <- lapply(mfd_list, function(x) x[rows1, "NO2"])
mfd_list_y2 <- lapply(mfd_list, function(x) x[rows2, "NO2"])

Then, the function fof_pc_real_time applies the function fof_pc to each element in mfd_list_x1 and mfd_list_x2.

mod_fof_pc_real_time_list <- fof_pc_real_time(
  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.

cc_list_real_time <- regr_cc_fof_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)