funcharts: control_charts_sof_pc – R documentation

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control_charts_sof_pc

Control charts for monitoring multivariate functional covariates and a scalar response

Description

This function builds a data frame needed to plot control charts for monitoring a multivariate functional covariates based on multivariate functional principal component analysis (MFPCA) and a related scalar response variable using the scalar-on-function regression control chart, as proposed in Capezza et al. (2020).

In particular, this function provides:

* the Hotelling's T^2 control chart,

* the squared prediction error (SPE) control chart,

* the scalar regression control chart.

This function calls control_charts_pca for the control charts on the multivariate functional covariates and regr_cc_sof for the scalar regression control chart.

The training data have already been used to fit the model. A tuning data set can be provided that is used to estimate the control chart limits. A phase II data set contains the observations to be monitored with the built control charts.

Usage

control_charts_sof_pc(
  mod,
  y_test,
  mfdobj_x_test,
  mfdobj_x_tuning = NULL,
  alpha = list(T2 = 0.0125, spe = 0.0125, y = 0.025),
  limits = "standard",
  seed = 0,
  nfold = NULL,
  ncores = 1
)

Arguments

`mod`	A list obtained as output from `sof_pc`, i.e. a fitted scalar-on-function linear regression model.
`y_test`	A numeric vector containing the observations of the scalar response variable in the phase II data set.
`mfdobj_x_test`	An object of class `mfd` containing the phase II data set of the functional covariates observations.
`mfdobj_x_tuning`	An object of class `mfd` containing the tuning set of the multivariate functional data, used to estimate the T^2 and SPE control chart limits. If NULL, the training data, i.e. the data used to fit the MFPCA model, are also used as the tuning data set, i.e. `tuning_data=pca$data`. Default is NULL.
`alpha`	A named list with three elements, named `T2`, `spe`, and codey, respectively, each containing the desired Type I error probability of the corresponding control chart (`T2` corresponds to the T^2 control chart, `spe` corresponds to the SPE control chart, `y` corresponds to the scalar regression control chart). Note that at the moment you have to take into account manually the family-wise error rate and adjust the two values accordingly. See Capezza et al. (2020) for additional details. Default value is `list(T2 = 0.0125, spe = 0.0125, y = 0.025)`.
`limits`	A character value. If "standard", it estimates the control limits on the tuning data set. If "cv", the function calculates the control limits only on the training data using cross-validation using `calculate_cv_limits`. Default is "standard".
`seed`	If `limits=="cv"`, since the split in the k groups is random, you can fix a seed to ensure reproducibility. Otherwise, this argument is ignored.
`nfold`	If `limits=="cv"`, this gives the number of groups k used for k-fold cross-validation. If it is equal to the number of observations in the training data set, then we have leave-one-out cross-validation. Otherwise, this argument is ignored.
`ncores`	If `limits=="cv"`, if you want perform the analysis in the k groups in parallel, give the number of cores/threads. Otherwise, this argument is ignored.

Value

A data.frame with as many rows as the number of multivariate functional observations in the phase II data set and the following columns:

* one id column identifying the multivariate functional observation in the phase II data set,

* one T2 column containing the Hotelling T^2 statistic calculated for all observations,

* one column per each functional variable, containing its contribution to the T^2 statistic,

* one spe column containing the SPE statistic calculated for all observations,

* one column per each functional variable, containing its contribution to the SPE statistic,

* T2_lim gives the upper control limit of the Hotelling's T^2 control chart,

* one contribution_T2_*_lim column per each functional variable giving the limits of the contribution of that variable to the Hotelling's T^2 statistic,

* spe_lim gives the upper control limit of the SPE control chart

* one contribution_spe*_lim column per each functional variable giving the limits of the contribution of that variable to the SPE statistic.

* y_hat: the predictions of the response variable corresponding to mfdobj_x_new,

* y: the same as the argument y_new given as input to this function,

* lwr: lower limit of the 1-alpha prediction interval on the response,

* pred_err: prediction error calculated as y-y_hat,

* pred_err_sup: upper limit of the 1-alpha prediction interval on the prediction error,

* pred_err_inf: lower limit of the 1-alpha prediction interval on the prediction error.

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[201:300, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates],
                         n_basis = 15,
                         lambda = 1e-2)
y <- rowMeans(air$NO2)
y1 <- y[1:60]
y2 <- y[91:100]
mfdobj_x1 <- mfdobj_x[1:60]
mfdobj_x_tuning <- mfdobj_x[61:90]
mfdobj_x2 <- mfdobj_x[91:100]
mod <- sof_pc(y1, mfdobj_x1)
cclist <- control_charts_sof_pc(mod = mod,
                                y_test = y2,
                                mfdobj_x_test = mfdobj_x2,
                                mfdobj_x_tuning = mfdobj_x_tuning)
plot_control_charts(cclist)

funcharts

Functional Control Charts

v1.0.0

GPL-3

Authors

Christian Capezza [cre, aut], Fabio Centofanti [aut], Antonio Lepore [aut], Alessandra Menafoglio [aut], Biagio Palumbo [aut], Simone Vantini [aut]

Initial release