In this vignette we show how to use the
package to apply the methods proposed in Capezza et al. (2020) to build
control charts for monitoring scalar quality characteristic adjusted for
by the effect of multivariate functional covariates, based on
scalar-on-function regression. 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. In order to show how the package
works, we consider a scalar-on-function regression model, where we take
the mean of
NO2 at each observation as the scalar response
and all other functions as functional covariates.
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
Then, we extract the scalar response variable, i.e. the mean of
NO2 at each observation:
y <- rowMeans(air$NO2)
In order to perform the statistical process monitoring analysis, we divide the data set into a phase I and a phase II dataset.
rows1 <- 1:300 rows2 <- 301:355 mfdobj_x1 <- mfdobj_x[rows1] mfdobj_x2 <- mfdobj_x[rows2] y1 <- y[rows1] y2 <- y[rows2]
We can build a scalar-on-function linear regression model where the
response variable is a linear function of the multivariate functional
principal components scores. The components to retain in the model can
be selected with
selection argument. Three alternatives are
available (default is
single_min_variance_explained.This criterion is used in Capezza et al. (2020).
mod <- sof_pc(y = y1, mfdobj_x = mfdobj_x1, selection = "PRESS", single_min_variance_explained = .01)
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 estimated regression model. It
is possible to plot the estimated functional regression coefficients,
which is also a multivariate functional data object of class
Moreover bootstrap can be used to obtain uncertainty quantification:
plot_bootstrap_sof_pc(mod, nboot = 10)
We can build control charts to monitor both functional covariates and the scalar response. We can either monitor the functional covariates alone or the functional covariates and the scalar response together, as performed in Capezza et al. (2020).
control_charts_pca provides a data frame
with all the information required to plot the desired control charts on
multivariate functional data, based on the Hotelling’s T^2 and the
squared prediction error statistics (see Capezza et al. (2020) and
Centofanti et al. (2021)). Among the arguments, you can pass a
tuning_data set, that 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 argument
contains the phase II data set of observations of multivariate
functional covariates, that are to be monitored with the control charts.
plot_control_charts returns a ggplot with the
cclist_pca <- control_charts_pca(pca = mod$pca, components = mod$components, newdata = mfdobj_x2) plot_control_charts(cclist_pca)
control_charts_sof_pc provides a data frame
with all the information required to plot the desired control charts. In
addition to the Hotelling’s T^2 and the squared prediction error
statistics obtained using
control_charts_pca (see previous
section), this function provides the scalar regression control charts,
which also monitors the prediction error on the response variable. The
plot_control_charts can be used to obtain
ggplot with the three control charts. As for monitoring the functional
covariates, you can pass a
tuning_data set (see previous
cclist_sof_pc <- control_charts_sof_pc(mod = mod, y_test = y2, mfdobj_x_test = mfdobj_x2) plot_control_charts(cclist_sof_pc)
Once an out-of-control observation is identified in a control chart, one can obtain the contribution of the functional covariates to the monitoring statistics to understand which variable was possibly responsible of the anomalous observation.
We can get the out of control observations using
ooc_index <- which_ooc(cclist_sof_pc) ooc_index #> $T2 #> n id #> 1 5 305 #> 2 12 312 #> #> $spe #> n id #> 1 9 309 #> #> $y #> n id #> 1 29 329 #> 2 52 352 #> 3 53 353 #> 4 54 354 #> 5 55 355
From the control charts it is possible to see some phase II
observations out of control. For example, let us consider observation 9.
We can exploit the contribution plots using the function
cont_plot to identify the variables responsible of the
anomaly. The contributions are plotted as bar plots, against upper
control limits estimated empirically on the training data.
Finally, we can also plot any multivariate functional observation
against a reference data set by using the function
plot_mon(cclist_sof_pc, mfdobj_x1, mfdobj_x2)
This helps to understand which part of the observations were anomalous. Variables exceeding contribution plot limits are coloured in red.
In Capezza et al. (2020), we have also implemented real-time
monitoring. 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[fun_covariates], grid = 1:24, n_basis = 15, lambda = 1e-2, k_seq = seq(0.5, 1, length.out = 7)) mfd_list1 <- lapply(mfd_list, function(x) x[rows1]) mfd_list2 <- lapply(mfd_list, function(x) x[rows2])
Then, the function
sof_pc_real_time applies the function
sof_pc to each element in
mod_list <- sof_pc_real_time(y = y1, mfd_real_time_list = mfd_list1)
Then, we can use
control_charts_sof_pc to each element in
mod_list and produce control charts for the phase II data
cc_list_real_time <- control_charts_sof_pc_real_time( mod_list = mod_list, y_test = y2, mfdobj_x_test = mfd_list2 )
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 9).
plot_control_charts_real_time(cc_list_real_time, id_num = 9)