funcharts: control_charts_pca – R documentation

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control_charts_pca

T^2 and SPE control charts for multivariate functional data

Description

This function builds a data frame needed to plot the Hotelling's T^2 and squared prediction error (SPE) control charts based on multivariate functional principal component analysis (MFPCA) performed on multivariate functional data, proposed in Capezza et al. (2020) together with the scalar control chart and used also to build the functional regression control chart proposed in Centofanti et al. (2020) (this function is used by regr_cc_fof). 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_pca(
  pca,
  components,
  tuning_data = NULL,
  newdata,
  alpha = list(T2 = 0.025, spe = 0.025),
  limits = "standard",
  seed = 0,
  nfold = 5,
  ncores = 1
)

Arguments

`pca`	An object of class `pca_mfd` obtained by doing MFPCA on the training set of multivariate functional data.
`components`	A vector of integers with the components over which to project the multivariate functional data.
`tuning_data`	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.
`newdata`	An object of class `mfd` containing the phase II set of the multivariate functional data to be monitored.
`alpha`	A named list with two elements, named `T2` and `spe`, respectively, each containing the desired Type I error probability of the corresponding 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) and Centofanti et al. (2020) for additional details. Default value is `list(T2 = 0.025, spe = 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.

References

Capezza C, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Control charts for monitoring ship operating conditions and CO2 emissions based on scalar-on-function regression. Applied Stochastic Models in Business and Industry, 36(3):477–500. <doi:10.1002/asmb.2507>

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Functional Regression Control Chart. Technometrics. <doi:10.1080/00401706.2020.1753581>

library(funcharts) data("air") air <- lapply(air, function(x) x[1:220, , drop = FALSE]) fun_covariates <- c("CO", "temperature") mfdobj_x <- get_mfd_list(air[fun_covariates], n_basis = 15, lambda = 1e-2) y <- rowMeans(air$NO2) y1 <- y[1:100] y_tuning <- y[101:200] y2 <- y[201:220] mfdobj_x1 <- mfdobj_x[1:100] mfdobj_x_tuning <- mfdobj_x[101:200] mfdobj_x2 <- mfdobj_x[201:220] pca <- pca_mfd(mfdobj_x1) components <- 1:which(cumsum(pca$varprop) >= .90)[1] cclist <- control_charts_pca(pca = pca, components = components, tuning_data = mfdobj_x_tuning, newdata = mfdobj_x2) plot_control_charts(cclist)

control_charts_pca

Description

Usage

Arguments

Value

References

See Also

funcharts

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