Scalar-on-function regression based on principal components

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 variance):

• if “variance”, the first M multivariate functional principal components are retained into the MFPCA model such that together they explain a fraction of variance greater than tot_variance_explained,
• if “PRESS”, each j-th functional principal component is retained into the MFPCA model if, by adding it to the set of the first j-1 functional principal components, then the predicted residual error sum of squares (PRESS) statistic decreases, and at the same time the fraction of variance explained by that single component is greater than single_min_variance_explained. This criterion is used in Capezza et al. (2020).
• if “gcv”, the criterion is equal as in the previous “PRESS” case, but the “PRESS” statistic is substituted by the generalized cross-validation (GCV) score.

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 mfd:

plot_mfd(mod$beta)

Moreover bootstrap can be used to obtain uncertainty quantification:

plot_bootstrap_sof_pc(mod, nboot = 10)

Control charts

We can build control charts to monitor both functional covariates and the scalar response. We can either monitor the functional covariates alone (unsupervised monitoring) or the functional covariates and the scalar response together (supervised monitoring), as performed in Capezza et al. (2020).

cclist_pca <- control_charts_pca(pca = mod$pca,
                                 components = mod$components,
                                 newdata = mfdobj_x2)
plot_control_charts(cclist_pca)

cclist_sof_pc <- control_charts_sof_pc(mod = mod,
                                       y_test = y2,
                                       mfdobj_x_test = mfdobj_x2)
plot_control_charts(cclist_sof_pc)

Fault detection

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 which_ooc. In the following, we focus on the supervised monitoring.

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.

cont_plot(cclist_sof_pc, 9)

Finally, we can also plot any multivariate functional observation against a reference data set by using the function plot_mon:

plot_mon(cclist_sof_pc, mfdobj_x1, mfdobj_x2[9])

This helps to understand which part of the observations were anomalous. Variables exceeding contribution plot limits are coloured in red.

Real-time monitoring

In Capezza et al. (2020), we have also implemented real-time monitoring (both unsupervised and supervised). 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 mfd_list1.

mod_list <- sof_pc_real_time(y = y1, mfd_real_time_list = mfd_list1)

Then, we can use control_charts_sof_pc_real_time to apply control_charts_sof_pc to each element in mod_list and produce control charts for the phase II data mfd_list2 and y2.

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)
#> Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> = "none")` instead.

#> Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> = "none")` instead.

#> Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> = "none")` instead.