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
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_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
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_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_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)
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
<- 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
<- 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)