Nothing
R/04_phaseII.R
In funcharts: Functional Control Charts
Defines functions
regr_cc_fof
regr_cc_sof
plot_control_charts
control_charts_sof_pc
control_charts_pca
Documented in
control_charts_pca
control_charts_sof_pc
plot_control_charts
regr_cc_fof
regr_cc_sof
#' T2 and SPE control charts for multivariate functional data
#'
#' This function builds a data frame needed to plot
#' the Hotelling's T2 and squared prediction error (SPE)
#' control charts
#' based on multivariate functional principal component analysis
#' (MFPCA) performed
#' on multivariate functional data, as Capezza et al. (2020)
#' for the multivariate functional covariates.
#' The training data have already been used to fit the model.
#' An optional tuning data set can be provided to estimate
#' the control chart limits.
#' A phase II data set contains the observations
#' to be monitored with the control charts.
#'
#' @param pca
#' An object of class \code{pca_mfd}
#' obtained by doing MFPCA on the
#' training set of multivariate functional data.
#' @param components
#' A vector of integers with the components over which
#' to project the multivariate functional data.
#' If this is not NULL, the arguments `single_min_variance_explained`
#' and `tot_variance_explained` are ignored.
#' If NULL, components are selected such that
#' the total fraction of variance explained by them
#' is at least equal to the argument `tot_variance_explained`,
#' where only components explaining individually a fraction of variance
#' at least equal to the argument `single_min_variance_explained`
#' are considered to be retained.
#' Default is NULL.
#' @param tuning_data
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' T2 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. \code{tuning_data=pca$data}.
#' Default is NULL.
#' @param newdata
#' An object of class \code{mfd} containing
#' the phase II set of the multivariate functional data to be monitored.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability and the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the two control charts equal to \code{alpha/2}.
#' If you want to set manually the Type-I error probabilities in the
#' two control charts, then the argument \code{alpha} must be
#' a named list
#' with two elements, named \code{T2} and \code{spe},
#' respectively, each containing
#' the desired Type I error probability of
#' the corresponding control chart.
#' Default value is 0.05.
#' @param 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 \code{calculate_cv_limits}. Default is "standard".
#' @param seed
#' If \code{limits=="cv"},
#' since the split in the k groups is random,
#' you can fix a seed to ensure reproducibility.
#' Deprecated: use \code{set.seed()} before calling
#' the function for reproducibility.
#' @param nfold
#' If \code{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.
#' @param ncores
#' If \code{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.
#' @param single_min_variance_explained
#' The minimum fraction of variance
#' that has to be explained
#' by each multivariate functional principal component
#' such that it is retained into the MFPCA model.
#' Default is 0.
#' @param tot_variance_explained
#' The minimum fraction of variance
#' that has to be explained
#' by the set of multivariate functional principal components
#' retained into the MFPCA model
#' fitted on the functional covariates.
#' Default is 0.9.
#' @param absolute_error
#' If FALSE, the SPE statistic, which monitors the principal components
#' not retained in the MFPCA model, is calculated as the sum
#' of the integrals of the squared prediction error functions, obtained
#' as the difference between the actual functions and their approximation
#' after projection over the selected principal components.
#' If TRUE, the SPE statistic is calculated by replacing the square of
#' the prediction errors with the absolute value, as proposed by
#' Capizzi and Masarotto (2018).
#' Default value is FALSE.
#'
#' @return
#' A \code{data.frame} with as many rows as the number of
#' multivariate functional observations in the phase II data set and
#' the following columns:
#'
#' * one \code{id} column identifying the multivariate functional observation
#' in the phase II data set,
#'
#' * one \code{T2} column containing the Hotelling T2 statistic
#' calculated for all observations,
#'
#' * one column per each functional variable,
#' containing its contribution to the T2 statistic,
#'
#' * one \code{spe} column containing the SPE statistic calculated
#' for all observations,
#'
#' * one column per each functional variable,
#' containing its contribution to the SPE statistic,
#'
#' * \code{T2_lim} gives the upper control limit of
#' the Hotelling's T2 control chart,
#'
#' * one \code{contribution_T2_*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable
#' to the Hotelling's T2 statistic,
#'
#' * \code{spe_lim} gives the upper control limit of the SPE control chart
#'
#' * one \code{contribution_spe*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable to the SPE statistic.
#'
#' @export
#'
#' @seealso \code{\link{regr_cc_fof}}
#'
#' @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.
#' \emph{Applied Stochastic Models in Business and Industry},
#' 36(3):477--500.
#' <doi:10.1002/asmb.2507>
#'
#' Capizzi, G., & Masarotto, G. (2018).
#' Phase I distribution-free analysis with the R package dfphase1.
#' In \emph{Frontiers in Statistical Quality Control 12 (pp. 3-19)}.
#' Springer International Publishing.
#'
#'
#' @examples
#' 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)
#' cclist <- control_charts_pca(pca = pca,
#' tuning_data = mfdobj_x_tuning,
#' newdata = mfdobj_x2)
#' plot_control_charts(cclist)
#'
control_charts_pca <- function(pca,
components = NULL,
tuning_data = NULL,
newdata,
alpha = 0.05,
limits = "standard",
seed,
nfold = 5,
ncores = 1,
tot_variance_explained = 0.9,
single_min_variance_explained = 0,
absolute_error = FALSE) {
if (!missing(seed)) {
warning(paste0("argument seed is deprecated; ",
"please use set.seed()
before calling the function instead."),
call. = FALSE)
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.list(alpha) & !all((names(alpha) %in% c("T2", "spe")))) {
stop("if alpha is a list, it must be named with names T2 and spe.")
}
if (!is.list(pca)) {
stop("pca must be a list produced by pca_mfd.")
}
if (length(limits) != 1) {
stop("Only one type of 'limits' allowed.")
}
if (!(limits %in% c("standard", "cv"))) {
stop("'limits' argument can be only 'standard' or 'cv'.")
}
if (is.null(components)) {
components_enough_var <- cumsum(pca$varprop) > tot_variance_explained
if (sum(components_enough_var) == 0)
ncomponents <- length(pca$varprop) else
ncomponents <- which(cumsum(pca$varprop) > tot_variance_explained)[1]
components <- seq_len(ncomponents)
components <-
which(pca$varprop[components] > single_min_variance_explained)
}
if (is.numeric(alpha)) alpha <- list(T2 = alpha / 2, spe = alpha / 2)
if (limits == "standard") lim <- calculate_limits(
pca = pca,
tuning_data = tuning_data,
components = components,
alpha = alpha,
absolute_error = absolute_error)
if (limits == "cv") lim <- calculate_cv_limits(
pca = pca,
components = components,
alpha = alpha,
nfold = nfold,
ncores = ncores,
absolute_error = absolute_error)
newdata_scaled <- scale_mfd(newdata,
center = pca$center_fd,
scale = if (pca$scale) pca$scale_fd else FALSE)
T2_spe <- get_T2_spe(pca,
components,
newdata_scaled = newdata_scaled,
absolute_error = absolute_error)
id <- data.frame(id = newdata$fdnames[[2]])
cbind(id, T2_spe, lim)
}
#' Control charts for monitoring a scalar quality characteristic adjusted for
#' by the effect of multivariate functional covariates
#'
#' @description
#' This function builds a data frame needed to
#' plot control charts
#' for monitoring a monitoring a scalar quality characteristic adjusted for
#' the effect of multivariate functional covariates based on
#' scalar-on-function regression,
#' as proposed in Capezza et al. (2020).
#'
#' In particular, this function provides:
#'
#' * the Hotelling's T2 control chart,
#'
#' * the squared prediction error (SPE) control chart,
#'
#' * the scalar regression control chart.
#'
#' This function calls \code{control_charts_pca} for the control charts on
#' the multivariate functional covariates and \code{\link{regr_cc_sof}}
#' for the scalar regression control chart.
#'
#' The training data have already been used to fit the model.
#' An optional 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 control charts.
#'
#'
#' @param mod
#' A list obtained as output from \code{sof_pc},
#' i.e. a fitted scalar-on-function linear regression model.
#' @param y_test
#' A numeric vector containing the observations
#' of the scalar response variable
#' in the phase II data set.
#' @param mfdobj_x_test
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates observations.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' T2 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. \code{tuning_data=pca$data}.
#' Default is NULL.
#' @param alpha
#' A named list with three elements, named \code{T2}, \code{spe},
#' and code{y},
#' respectively, each containing
#' the desired Type I error probability of the corresponding control chart
#' (\code{T2} corresponds to the T2 control chart,
#' \code{spe} corresponds to the SPE control chart,
#' \code{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
#' \code{list(T2 = 0.0125, spe = 0.0125, y = 0.025)}.
#' @param 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 \code{calculate_cv_limits}. Default is "standard".
#' @param seed
#' If \code{limits=="cv"},
#' since the split in the k groups is random,
#' you can fix a seed to ensure reproducibility.
#' Deprecated: use \code{set.seed()} before calling
#' the function for reproducibility.
#' @param nfold
#' If \code{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.
#' @param ncores
#' If \code{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.
#'
#' @return
#' A \code{data.frame} with as many rows as the number of
#' multivariate functional observations in the phase II data set and
#' the following columns:
#'
#' * one \code{id} column identifying the multivariate functional observation
#' in the phase II data set,
#'
#' * one \code{T2} column containing the Hotelling T2 statistic calculated
#' for all observations,
#'
#' * one column per each functional variable, containing its contribution
#' to the T2 statistic,
#'
#' * one \code{spe} column containing the SPE statistic calculated
#' for all observations,
#'
#' * one column per each functional variable, containing its contribution
#' to the SPE statistic,
#'
#' * \code{T2_lim} gives the upper control limit of the
#' Hotelling's T2 control chart,
#'
#' * one \code{contribution_T2_*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable to the
#' Hotelling's T2 statistic,
#'
#' * \code{spe_lim} gives the upper control limit of the SPE control chart
#'
#' * one \code{contribution_spe*_lim} column per
#' each functional variable giving the
#' limits of the contribution of that variable to the SPE statistic.
#'
#' * \code{y_hat}: the predictions of the response variable
#' corresponding to \code{mfdobj_x_new},
#'
#' * \code{y}: the same as the argument \code{y_new}
#' given as input to this function,
#'
#' * \code{lwr}: lower limit of the \code{1-alpha}
#' prediction interval on the response,
#'
#' * \code{pred_err}: prediction error calculated as \code{y-y_hat},
#'
#' * \code{pred_err_sup}: upper limit of the \code{1-alpha}
#' prediction interval on the prediction error,
#'
#' * \code{pred_err_inf}: lower limit of the \code{1-alpha}
#' prediction interval on the prediction error.
#'
#' @export
#'
#' @seealso \code{\link{control_charts_pca}}, \code{\link{regr_cc_sof}}
#'
#' @examples
#' \dontrun{
#' #' 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)
#' }
#'
control_charts_sof_pc <- function(mod,
y_test,
mfdobj_x_test,
mfdobj_x_tuning = NULL,
alpha = list(
T2 = .0125,
spe = .0125,
y = .025),
limits = "standard",
seed,
nfold = NULL,
ncores = 1) {
.Deprecated("regr_cc_sof")
regr_cc_sof(object = mod,
y_new = y_test,
mfdobj_x_new = mfdobj_x_test,
mfdobj_x_tuning = mfdobj_x_tuning,
y_tuning = NULL,
alpha = alpha,
parametric_limits = TRUE,
include_covariates = TRUE)
}
#' Plot control charts
#'
#' This function takes as input a data frame produced
#' with functions such as
#' \code{\link{control_charts_pca}} and \code{\link{control_charts_sof_pc}} and
#' produces a ggplot with the desired control charts, i.e.
#' it plots a point for each
#' observation in the phase II data set against
#' the corresponding control limits.
#'
#' @param cclist
#' A \code{data.frame} produced by
#' \code{\link{control_charts_pca}}, \code{\link{control_charts_sof_pc}}
#' \code{\link{regr_cc_fof}}, or \code{\link{regr_cc_sof}}.
#' @param nobsI
#' An integer indicating the first observations that are plotted in gray.
#' It is useful when one wants to plot the phase I data set together
#' with the phase II data. In that case, one needs to indicate the number
#' of phase I observations included in \code{cclist}.
#' Default is zero.
#'
#' @return A ggplot with the functional control charts.
#'
#' @details Out-of-control points are signaled by colouring them in red.
#'
#' @export
#' @examples
#' library(funcharts)
#' data("air")
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y <- get_mfd_list(air["NO2"],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y1 <- mfdobj_y[1:60]
#' mfdobj_y_tuning <- mfdobj_y[61:90]
#' mfdobj_y2 <- mfdobj_y[91:100]
#' mfdobj_x1 <- mfdobj_x[1:60]
#' mfdobj_x_tuning <- mfdobj_x[61:90]
#' mfdobj_x2 <- mfdobj_x[91:100]
#' mod_fof <- fof_pc(mfdobj_y1, mfdobj_x1)
#' cclist <- regr_cc_fof(mod_fof,
#' mfdobj_y_new = mfdobj_y2,
#' mfdobj_x_new = mfdobj_x2,
#' mfdobj_y_tuning = NULL,
#' mfdobj_x_tuning = NULL)
#' plot_control_charts(cclist)
#'
plot_control_charts <- function(cclist, nobsI = 0) {
if (!is.data.frame(cclist)) {
stop(paste0("cclist must be a data frame ",
"containing data to produce control charts."))
}
df_hot <- NULL
if (!is.null(cclist$T2)) {
df_hot <- data.frame(statistic = cclist$T2,
lcl = 0,
ucl = cclist$T2_lim) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "HOTELLING~T2~CONTROL~CHART")
hot_range <- df_hot %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_hot <- df_hot %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - hot_range * 0.2,
statistic > ucl ~ statistic + hot_range * 0.2,
TRUE ~ statistic,
))
}
df_spe <- NULL
if (!is.null(cclist$spe)) {
df_spe <- data.frame(statistic = cclist$spe,
lcl = 0,
ucl = cclist$spe_lim) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "SPE~CONTROL~CHART")
spe_range <- df_spe %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_spe <- df_spe %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - spe_range * 0.2,
statistic > ucl ~ statistic + spe_range * 0.2,
TRUE ~ statistic,
))
}
df_y <- NULL
if (!is.null(cclist$pred_err)) {
df_y <- data.frame(statistic = cclist$pred_err,
lcl = cclist$pred_err_inf,
ucl = cclist$pred_err_sup)
y_range <- df_y %>%
select(c(.data$statistic, .data$ucl, .data$lcl)) %>%
range() %>%
diff()
df_y <- df_y %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - y_range * 0.2,
statistic > ucl ~ statistic + y_range * 0.2,
TRUE ~ statistic,
))
df_y <- df_y %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl | .data$statistic < .data$lcl,
type = "REGRESSION~CONTROL~CHART")
}
df_hot_x <- NULL
if (!is.null(cclist$T2_x)) {
df_hot_x <- data.frame(statistic = cclist$T2_x,
lcl = 0,
ucl = cclist$T2_lim_x) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "HOTELLING~T2~CONTROL~CHART~(COVARIATES)")
hot_range <- df_hot_x %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_hot_x <- df_hot_x %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - hot_range * 0.2,
statistic > ucl ~ statistic + hot_range * 0.2,
TRUE ~ statistic,
))
}
df_spe_x <- NULL
if (!is.null(cclist$spe_x)) {
df_spe_x <- data.frame(statistic = cclist$spe_x,
lcl = 0,
ucl = cclist$spe_lim_x) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "SPE~CONTROL~CHART~(COVARIATES)")
spe_range <- df_spe_x %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_spe_x <- df_spe_x %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - spe_range * 0.2,
statistic > ucl ~ statistic + spe_range * 0.2,
TRUE ~ statistic,
))
}
plot_list <- list()
if (!is.null(cclist$T2)) {
plot_list$p_hot <- ggplot(df_hot, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_hot, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(T2~statistic)) +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART))
}
if (!is.null(cclist$spe)) {
plot_list$p_spe <- ggplot(df_spe, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_spe, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(SPE~statistic)) +
ggtitle(expression(SPE~CONTROL~CHART))
}
if (!is.null(cclist$pred_err)) {
plot_list$p_y <- ggplot(df_y, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_line(aes(y = .data$lcl), lty = 2) +
geom_line(aes(y = .data$ucl), lty = 2) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_y, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression("Regression residuals")) +
ggtitle(expression(REGRESSION~CONTROL~CHART))
}
if (!is.null(cclist$T2_x)) {
plot_list$p_hot_x <- ggplot(df_hot_x, aes(x = .data$id,
y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_hot_x, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(T2~statistic)) +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART~(COVARIATES)))
plot_list$p_hot <- plot_list$p_hot +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART~(RESPONSE)))
}
if (!is.null(cclist$spe_x)) {
plot_list$p_spe_x <- ggplot(df_spe_x, aes(x = .data$id,
y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_spe_x, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(SPE~statistic)) +
ggtitle(expression(SPE~CONTROL~CHART~(COVARIATES)))
plot_list$p_spe <- plot_list$p_spe +
ggtitle(expression(SPE~CONTROL~CHART~(RESPONSE)))
}
p <- patchwork::wrap_plots(plot_list, ncol = 1) &
scale_color_manual(values = c("TRUE" = "red",
"FALSE" = "black",
"phase1" = "grey")) &
scale_x_continuous(
limits = c(0, nrow(cclist) + 1),
breaks = seq(1, nrow(cclist), by = round(nrow(cclist) / 50) + 1),
expand = c(0.015, 0.015))
if (nobsI > 0) {
nplots <- length(p$patches$plots) + 1
for (jj in seq_len(nplots)) {
p[[jj]]$data$ooc[seq_len(nobsI)] <- "phase1"
p[[jj]]$layers[[5]]$data <- p[[jj]]$layers[[5]]$data %>%
filter(.data$id > nobsI) %>%
as.data.frame()
p[[jj]] <- p[[jj]] +
geom_point(aes_string(y = "ucl"),
pch = "-",
size = 5,
col = "grey",
data = filter(p[[jj]]$data, .data$id <= nobsI)) +
geom_line(aes_string("id", "statistic"),
col = "grey",
data = filter(p[[jj]]$data, .data$id < nobsI))
}
p <- p &
geom_vline(aes(xintercept = nobsI + 1), lty = 2)
}
return(p)
}
#' Scalar-on-Function Regression Control Chart
#'
#' This function is deprecated. Use \code{\link{regr_cc_sof}}.
#' This function builds a data frame needed
#' to plot the scalar-on-function regression control chart,
#' based on a fitted function-on-function linear regression model and
#' proposed in Capezza et al. (2020).
#' If \code{include_covariates} is \code{TRUE},
#' it also plots the Hotelling's T2 and
#' squared prediction error control charts built on the
#' multivariate functional covariates.
#'
#' The training data have already been used to fit the model.
#' An additional 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.
#'
#' @param object
#' A list obtained as output from \code{sof_pc},
#' i.e. a fitted scalar-on-function linear regression model.
#' @param mfdobj_x_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates observations.
#' @param y_new
#' A numeric vector containing the observations of
#' the scalar response variable
#' in the phase II data set.
#' @param y_tuning
#' A numeric vector containing the observations of the scalar response
#' variable in the tuning data set.
#' If NULL, the training data, i.e. the data used to
#' fit the scalar-on-function regression model,
#' are also used as the tuning data set.
#' Default is NULL.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' control chart limits.
#' If NULL, the training data, i.e. the data used to
#' fit the scalar-on-function regression model,
#' are also used as the tuning data set.
#' Default is NULL.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability.
#' If \code{include_covariates} is \code{TRUE}, i.e.,
#' also the Hotelling's T2 and SPE control charts are built
#' on the functional covariates, then the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the three control charts equal to \code{alpha/3}.
#' In this last case,
#' if you want to set manually the Type-I error probabilities,
#' then the argument \code{alpha} must be a named list
#' with three elements, named \code{T2}, \code{spe} and \code{y},
#' respectively, each containing
#' the desired Type I error probability of
#' the corresponding control chart, where \code{y} refers to the
#' regression control chart.
#' Default value is 0.05.
#' @param parametric_limits
#' If \code{TRUE}, the limits are calculated based on the normal distribution
#' assumption on the response variable, as in Capezza et al. (2020).
#' If \code{FALSE}, the limits are calculated nonparametrically as
#' empirical quantiles of the distribution of the residuals calculated
#' on the tuning data set.
#' The default value is \code{FALSE}.
#' @param include_covariates
#' If TRUE, also functional covariates are monitored through
#' \code{control_charts_pca},.
#' If FALSE, only the scalar response, conditionally on the covariates,
#' is monitored.
#' @param absolute_error
#' A logical value that, if \code{include_covariates} is TRUE, is passed
#' to \code{\link{control_charts_pca}}.
#'
#' @return
#' A \code{data.frame} with as many rows as the
#' number of functional replications in \code{mfdobj_x_new},
#' with the following columns:
#'
#' * \code{y_hat}: the predictions of the response variable
#' corresponding to \code{mfdobj_x_new},
#'
#' * \code{y}: the same as the argument \code{y_new} given as input
#' to this function,
#'
#' * \code{lwr}: lower limit of the \code{1-alpha} prediction interval
#' on the response,
#'
#' * \code{pred_err}: prediction error calculated as \code{y-y_hat},
#'
#' * \code{pred_err_sup}: upper limit of the \code{1-alpha} prediction interval
#' on the prediction error,
#'
#' * \code{pred_err_inf}: lower limit of the \code{1-alpha} prediction interval
#' on the prediction error.
#'
#' @export
#'
#' @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.
#' \emph{Applied Stochastic Models in Business and Industry},
#' 36(3):477--500.
#' <doi:10.1002/asmb.2507>
#'
#' @examples
#' library(funcharts)
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' y <- rowMeans(air$NO2)
#' y1 <- y[1:80]
#' y2 <- y[81:100]
#' mfdobj_x1 <- mfdobj_x[1:80]
#' mfdobj_x2 <- mfdobj_x[81:100]
#' mod <- sof_pc(y1, mfdobj_x1)
#' cclist <- regr_cc_sof(object = mod,
#' y_new = y2,
#' mfdobj_x_new = mfdobj_x2)
#' plot_control_charts(cclist)
#'
regr_cc_sof <- function(object,
y_new,
mfdobj_x_new,
y_tuning = NULL,
mfdobj_x_tuning = NULL,
alpha = 0.05,
parametric_limits = FALSE,
include_covariates = FALSE,
absolute_error = FALSE) {
if (!is.list(object)) {
stop("object must be a list produced by sof_pc.")
}
if (!identical(names(object), c(
"mod",
"pca",
"beta_fd",
"residuals",
"components",
"selection",
"single_min_variance_explained",
"tot_variance_explained",
"gcv",
"PRESS"
))) {
stop("object must be a list produced by sof_pc.")
}
if (!is.numeric(y_new)) {
stop("y_new must be numeric.")
}
if (!is.null(mfdobj_x_new)) {
if (!is.mfd(mfdobj_x_new)) {
stop("mfdobj_x_new must be an object from mfd class.")
}
if (dim(mfdobj_x_new$coefs)[3] != dim(object$pca$data$coefs)[3]) {
stop(paste0("mfdobj_x_new must have the same number of variables ",
"as training data."))
}
}
nobsx <- dim(mfdobj_x_new$coefs)[2]
nobsy <- length(y_new)
if (nobsx != nobsy) {
stop(paste0("y_new and mfdobj_x_new must have ",
"the same number of observations."))
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha)) {
if (include_covariates) {
alpha <- list(T2 = alpha / 3, spe = alpha / 3, y = alpha / 3)
} else {
alpha <- list(y = alpha)
}
}
if (is.list(alpha)) {
if (include_covariates) {
if (!all(names(alpha) %in% c("T2", "spe", "y")) |
!all(c("T2", "spe", "y") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is TRUE,
it must be named with names T2, spe, y.")
}
}
if (!include_covariates) {
if (!all(names(alpha) %in% "y") |
!all("y" %in% names(alpha))) {
stop("if alpha is a list and include_covariates is FALSE,
it must be named with name y.")
}
}
}
y_hat <- predict_sof_pc(object = object,
y_new = y_new,
mfdobj_x_new = mfdobj_x_new,
alpha = alpha$y)
ret <- data.frame(
y_hat,
pred_err_sup = y_hat$upr - y_hat$fit,
pred_err_inf = y_hat$lwr - y_hat$fit)
if (!parametric_limits) {
if (is.null(y_tuning) | is.null(mfdobj_x_tuning)) {
fml <- formula(object$mod)
response_name <- all.vars(fml)[1]
y_tuning <- object$mod$model[, response_name]
mfdobj_x_tuning <- object$pca$data
}
y_hat_tuning <- predict_sof_pc(object = object,
y_new = y_tuning,
mfdobj_x_new = mfdobj_x_tuning,
alpha = alpha$y)
upr_lim <- as.numeric(quantile(y_hat_tuning$pred_err, 1 - alpha$y/2))
lwr_lim <- as.numeric(quantile(y_hat_tuning$pred_err, alpha$y/2))
ret$pred_err_sup <- upr_lim
ret$pred_err_inf <- lwr_lim
}
if (include_covariates) {
alpha_x <- alpha[c("T2", "spe")]
ccpca <- control_charts_pca(pca = object$pca,
components = object$components,
tuning_data = mfdobj_x_tuning,
newdata = mfdobj_x_new,
alpha = alpha_x,
limits = "standard",
absolute_error = absolute_error)
ret <- cbind(ccpca, ret)
} else {
ret <- cbind(data.frame(id = mfdobj_x_new$fdnames[[2]]), ret)
}
return(ret)
}
#' Functional Regression Control Chart
#'
#' It builds a data frame needed to plot the
#' Functional Regression Control Chart
#' introduced in Centofanti et al. (2021),
#' for monitoring a functional quality characteristic adjusted for
#' by the effect of multivariate functional covariates,
#' based on a fitted
#' function-on-function linear regression model.
#' The training data have already been used to fit the model.
#' An optional 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 control charts.
#' It also allows to jointly monitor the multivariate functional covariates.
#'
#' @param object
#' A list obtained as output from \code{fof_pc},
#' i.e. a fitted function-on-function linear regression model.
#' @param mfdobj_y_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional response
#' observations to be monitored.
#' @param mfdobj_x_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates
#' observations to be monitored.
#' @param mfdobj_y_tuning
#' An object of class \code{mfd} containing
#' the tuning data set of the functional response observations,
#' used to estimate the control chart limits.
#' If NULL, the training data, i.e. the data used to fit the
#' function-on-function linear regression model,
#' are also used as the tuning data set, i.e.
#' \code{mfdobj_y_tuning=object$pca_y$data}.
#' Default is NULL.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning data set of the functional covariates observations,
#' used to estimate the control chart limits.
#' If NULL, the training data, i.e. the data used to fit the
#' function-on-function linear regression model,
#' are also used as the tuning data set, i.e.
#' \code{mfdobj_x_tuning=object$pca_x$data}.
#' Default is NULL.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability.
#' By default, it is equal to 0.05 and the Bonferroni correction
#' is applied by setting the type-I error probabilities equal to
#' \code{alpha/2} in the Hotelling's T2 and SPE control charts.
#' If \code{include_covariates} is \code{TRUE}, i.e.,
#' the Hotelling's T2 and SPE control charts are built
#' also on the multivariate functional covariates, then the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the four control charts equal to \code{alpha/4}.
#' If you want to set manually the Type-I error probabilities,
#' then the argument \code{alpha} must be a named list
#' with elements named as \code{T2}, \code{spe},
#' \code{T2_x} and, \code{spe_x}, respectively, containing
#' the desired Type I error probability of
#' the T2 and SPE control charts for the functional response and
#' the multivariate functional covariates, respectively.
#' @param include_covariates
#' If TRUE, also functional covariates are monitored through
#' \code{control_charts_pca},.
#' If FALSE, only the functional response, conditionally on the covariates,
#' is monitored.
#' @param absolute_error
#' A logical value that, if \code{include_covariates} is TRUE, is passed
#' to \code{\link{control_charts_pca}}.
#'
#' @return
#' A \code{data.frame} containing the output of the
#' function \code{control_charts_pca} applied to
#' the prediction errors.
#' @export
#'
#' @seealso \code{\link{control_charts_pca}}
#'
#' @references
#' Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021)
#' Functional Regression Control Chart.
#' \emph{Technometrics}, 63(3), 281--294. <doi:10.1080/00401706.2020.1753581>
#'
#' @examples
#' library(funcharts)
#' data("air")
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y <- get_mfd_list(air["NO2"],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y1 <- mfdobj_y[1:60]
#' mfdobj_y_tuning <- mfdobj_y[61:90]
#' mfdobj_y2 <- mfdobj_y[91:100]
#' mfdobj_x1 <- mfdobj_x[1:60]
#' mfdobj_x_tuning <- mfdobj_x[61:90]
#' mfdobj_x2 <- mfdobj_x[91:100]
#' mod_fof <- fof_pc(mfdobj_y1, mfdobj_x1)
#' cclist <- regr_cc_fof(mod_fof,
#' mfdobj_y_new = mfdobj_y2,
#' mfdobj_x_new = mfdobj_x2,
#' mfdobj_y_tuning = NULL,
#' mfdobj_x_tuning = NULL)
#' plot_control_charts(cclist)
#'
regr_cc_fof <- function(object,
mfdobj_y_new,
mfdobj_x_new,
mfdobj_y_tuning = NULL,
mfdobj_x_tuning = NULL,
alpha = 0.05,
include_covariates = FALSE,
absolute_error = FALSE) {
if (!is.list(object)) {
stop("object must be a list produced by fof_pc.")
}
if (!identical(names(object), c(
"mod",
"beta_fd",
"fitted.values",
"residuals_original_scale",
"residuals",
"type_residuals",
"pca_x",
"pca_y",
"pca_res",
"components_x",
"components_y",
"components_res",
"y_standardized",
"tot_variance_explained_x",
"tot_variance_explained_y",
"tot_variance_explained_res",
"get_studentized_residuals"
))) {
stop("object must be a list produced by fof_pc.")
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.list(alpha)) {
if (include_covariates) {
if (!all(names(alpha) %in% c("T2_x", "spe_x", "T2", "spe")) |
!all(c("T2_x", "spe_x", "T2", "spe") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is TRUE,
it must be named with names T2_x spe_x, T2, spe.")
}
}
if (!include_covariates) {
if (!all(names(alpha) %in% c("T2", "spe")) |
!all(c("T2", "spe") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is FALSE,
it must be named with names T2 and spe.")
}
}
alpha_x <- alpha[c("T2_x", "spe_x")]
alpha_y <- alpha[c("T2", "spe")]
}
if (is.numeric(alpha)) {
alpha_y <- list(T2 = alpha / 4, spe = alpha / 4)
if (include_covariates) {
alpha_x <- list(T2_x = alpha / 4, spe_x = alpha / 4)
}
}
if (is.null(mfdobj_y_tuning) | is.null(mfdobj_x_tuning)) {
mfdobj_y_tuning <- object$pca_y$data
mfdobj_x_tuning <- object$pca_x$data
}
tuning <- predict_fof_pc(
object = object,
mfdobj_y_new = mfdobj_y_tuning,
mfdobj_x_new = mfdobj_x_tuning)
phase_II <- predict_fof_pc(
object = object,
mfdobj_y_new = mfdobj_y_new,
mfdobj_x_new = mfdobj_x_new)
out <- control_charts_pca(
pca = object$pca_res,
components = object$components_res,
tuning_data = tuning$pred_error,
newdata = phase_II$pred_error,
alpha = alpha_y,
limits = "standard"
)
ret <- out %>%
select(-contains("contribution_"))
if (include_covariates) {
alpha_x <- list(T2 = alpha_x$T2, spe = alpha_x$spe)
ret_covariates <- control_charts_pca(
pca = object$pca_x,
components = object$components_x,
tuning_data = mfdobj_x_tuning,
newdata = mfdobj_x_new,
alpha = alpha_x,
limits = "standard",
absolute_error = absolute_error
) %>%
rename("T2_x" = "T2",
"spe_x" = "spe",
"T2_lim_x" = "T2_lim",
"spe_lim_x" = "spe_lim")
ret <- cbind(ret, select(ret_covariates, -"id"))
}
return(ret)
}
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Defines functions regr_cc_fof regr_cc_sof plot_control_charts control_charts_sof_pc control_charts_pca
Documented in control_charts_pca control_charts_sof_pc plot_control_charts regr_cc_fof regr_cc_sof
#' T2 and SPE control charts for multivariate functional data
#'
#' This function builds a data frame needed to plot
#' the Hotelling's T2 and squared prediction error (SPE)
#' control charts
#' based on multivariate functional principal component analysis
#' (MFPCA) performed
#' on multivariate functional data, as Capezza et al. (2020)
#' for the multivariate functional covariates.
#' The training data have already been used to fit the model.
#' An optional tuning data set can be provided to estimate
#' the control chart limits.
#' A phase II data set contains the observations
#' to be monitored with the control charts.
#'
#' @param pca
#' An object of class \code{pca_mfd}
#' obtained by doing MFPCA on the
#' training set of multivariate functional data.
#' @param components
#' A vector of integers with the components over which
#' to project the multivariate functional data.
#' If this is not NULL, the arguments `single_min_variance_explained`
#' and `tot_variance_explained` are ignored.
#' If NULL, components are selected such that
#' the total fraction of variance explained by them
#' is at least equal to the argument `tot_variance_explained`,
#' where only components explaining individually a fraction of variance
#' at least equal to the argument `single_min_variance_explained`
#' are considered to be retained.
#' Default is NULL.
#' @param tuning_data
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' T2 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. \code{tuning_data=pca$data}.
#' Default is NULL.
#' @param newdata
#' An object of class \code{mfd} containing
#' the phase II set of the multivariate functional data to be monitored.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability and the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the two control charts equal to \code{alpha/2}.
#' If you want to set manually the Type-I error probabilities in the
#' two control charts, then the argument \code{alpha} must be
#' a named list
#' with two elements, named \code{T2} and \code{spe},
#' respectively, each containing
#' the desired Type I error probability of
#' the corresponding control chart.
#' Default value is 0.05.
#' @param 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 \code{calculate_cv_limits}. Default is "standard".
#' @param seed
#' If \code{limits=="cv"},
#' since the split in the k groups is random,
#' you can fix a seed to ensure reproducibility.
#' Deprecated: use \code{set.seed()} before calling
#' the function for reproducibility.
#' @param nfold
#' If \code{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.
#' @param ncores
#' If \code{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.
#' @param single_min_variance_explained
#' The minimum fraction of variance
#' that has to be explained
#' by each multivariate functional principal component
#' such that it is retained into the MFPCA model.
#' Default is 0.
#' @param tot_variance_explained
#' The minimum fraction of variance
#' that has to be explained
#' by the set of multivariate functional principal components
#' retained into the MFPCA model
#' fitted on the functional covariates.
#' Default is 0.9.
#' @param absolute_error
#' If FALSE, the SPE statistic, which monitors the principal components
#' not retained in the MFPCA model, is calculated as the sum
#' of the integrals of the squared prediction error functions, obtained
#' as the difference between the actual functions and their approximation
#' after projection over the selected principal components.
#' If TRUE, the SPE statistic is calculated by replacing the square of
#' the prediction errors with the absolute value, as proposed by
#' Capizzi and Masarotto (2018).
#' Default value is FALSE.
#'
#' @return
#' A \code{data.frame} with as many rows as the number of
#' multivariate functional observations in the phase II data set and
#' the following columns:
#'
#' * one \code{id} column identifying the multivariate functional observation
#' in the phase II data set,
#'
#' * one \code{T2} column containing the Hotelling T2 statistic
#' calculated for all observations,
#'
#' * one column per each functional variable,
#' containing its contribution to the T2 statistic,
#'
#' * one \code{spe} column containing the SPE statistic calculated
#' for all observations,
#'
#' * one column per each functional variable,
#' containing its contribution to the SPE statistic,
#'
#' * \code{T2_lim} gives the upper control limit of
#' the Hotelling's T2 control chart,
#'
#' * one \code{contribution_T2_*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable
#' to the Hotelling's T2 statistic,
#'
#' * \code{spe_lim} gives the upper control limit of the SPE control chart
#'
#' * one \code{contribution_spe*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable to the SPE statistic.
#'
#' @export
#'
#' @seealso \code{\link{regr_cc_fof}}
#'
#' @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.
#' \emph{Applied Stochastic Models in Business and Industry},
#' 36(3):477--500.
#' <doi:10.1002/asmb.2507>
#'
#' Capizzi, G., & Masarotto, G. (2018).
#' Phase I distribution-free analysis with the R package dfphase1.
#' In \emph{Frontiers in Statistical Quality Control 12 (pp. 3-19)}.
#' Springer International Publishing.
#'
#'
#' @examples
#' 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)
#' cclist <- control_charts_pca(pca = pca,
#' tuning_data = mfdobj_x_tuning,
#' newdata = mfdobj_x2)
#' plot_control_charts(cclist)
#'
control_charts_pca <- function(pca,
components = NULL,
tuning_data = NULL,
newdata,
alpha = 0.05,
limits = "standard",
seed,
nfold = 5,
ncores = 1,
tot_variance_explained = 0.9,
single_min_variance_explained = 0,
absolute_error = FALSE) {
if (!missing(seed)) {
warning(paste0("argument seed is deprecated; ",
"please use set.seed()
before calling the function instead."),
call. = FALSE)
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.list(alpha) & !all((names(alpha) %in% c("T2", "spe")))) {
stop("if alpha is a list, it must be named with names T2 and spe.")
}
if (!is.list(pca)) {
stop("pca must be a list produced by pca_mfd.")
}
if (length(limits) != 1) {
stop("Only one type of 'limits' allowed.")
}
if (!(limits %in% c("standard", "cv"))) {
stop("'limits' argument can be only 'standard' or 'cv'.")
}
if (is.null(components)) {
components_enough_var <- cumsum(pca$varprop) > tot_variance_explained
if (sum(components_enough_var) == 0)
ncomponents <- length(pca$varprop) else
ncomponents <- which(cumsum(pca$varprop) > tot_variance_explained)[1]
components <- seq_len(ncomponents)
components <-
which(pca$varprop[components] > single_min_variance_explained)
}
if (is.numeric(alpha)) alpha <- list(T2 = alpha / 2, spe = alpha / 2)
if (limits == "standard") lim <- calculate_limits(
pca = pca,
tuning_data = tuning_data,
components = components,
alpha = alpha,
absolute_error = absolute_error)
if (limits == "cv") lim <- calculate_cv_limits(
pca = pca,
components = components,
alpha = alpha,
nfold = nfold,
ncores = ncores,
absolute_error = absolute_error)
newdata_scaled <- scale_mfd(newdata,
center = pca$center_fd,
scale = if (pca$scale) pca$scale_fd else FALSE)
T2_spe <- get_T2_spe(pca,
components,
newdata_scaled = newdata_scaled,
absolute_error = absolute_error)
id <- data.frame(id = newdata$fdnames[[2]])
cbind(id, T2_spe, lim)
}
#' Control charts for monitoring a scalar quality characteristic adjusted for
#' by the effect of multivariate functional covariates
#'
#' @description
#' This function builds a data frame needed to
#' plot control charts
#' for monitoring a monitoring a scalar quality characteristic adjusted for
#' the effect of multivariate functional covariates based on
#' scalar-on-function regression,
#' as proposed in Capezza et al. (2020).
#'
#' In particular, this function provides:
#'
#' * the Hotelling's T2 control chart,
#'
#' * the squared prediction error (SPE) control chart,
#'
#' * the scalar regression control chart.
#'
#' This function calls \code{control_charts_pca} for the control charts on
#' the multivariate functional covariates and \code{\link{regr_cc_sof}}
#' for the scalar regression control chart.
#'
#' The training data have already been used to fit the model.
#' An optional 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 control charts.
#'
#'
#' @param mod
#' A list obtained as output from \code{sof_pc},
#' i.e. a fitted scalar-on-function linear regression model.
#' @param y_test
#' A numeric vector containing the observations
#' of the scalar response variable
#' in the phase II data set.
#' @param mfdobj_x_test
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates observations.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' T2 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. \code{tuning_data=pca$data}.
#' Default is NULL.
#' @param alpha
#' A named list with three elements, named \code{T2}, \code{spe},
#' and code{y},
#' respectively, each containing
#' the desired Type I error probability of the corresponding control chart
#' (\code{T2} corresponds to the T2 control chart,
#' \code{spe} corresponds to the SPE control chart,
#' \code{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
#' \code{list(T2 = 0.0125, spe = 0.0125, y = 0.025)}.
#' @param 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 \code{calculate_cv_limits}. Default is "standard".
#' @param seed
#' If \code{limits=="cv"},
#' since the split in the k groups is random,
#' you can fix a seed to ensure reproducibility.
#' Deprecated: use \code{set.seed()} before calling
#' the function for reproducibility.
#' @param nfold
#' If \code{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.
#' @param ncores
#' If \code{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.
#'
#' @return
#' A \code{data.frame} with as many rows as the number of
#' multivariate functional observations in the phase II data set and
#' the following columns:
#'
#' * one \code{id} column identifying the multivariate functional observation
#' in the phase II data set,
#'
#' * one \code{T2} column containing the Hotelling T2 statistic calculated
#' for all observations,
#'
#' * one column per each functional variable, containing its contribution
#' to the T2 statistic,
#'
#' * one \code{spe} column containing the SPE statistic calculated
#' for all observations,
#'
#' * one column per each functional variable, containing its contribution
#' to the SPE statistic,
#'
#' * \code{T2_lim} gives the upper control limit of the
#' Hotelling's T2 control chart,
#'
#' * one \code{contribution_T2_*_lim} column per each
#' functional variable giving the
#' limits of the contribution of that variable to the
#' Hotelling's T2 statistic,
#'
#' * \code{spe_lim} gives the upper control limit of the SPE control chart
#'
#' * one \code{contribution_spe*_lim} column per
#' each functional variable giving the
#' limits of the contribution of that variable to the SPE statistic.
#'
#' * \code{y_hat}: the predictions of the response variable
#' corresponding to \code{mfdobj_x_new},
#'
#' * \code{y}: the same as the argument \code{y_new}
#' given as input to this function,
#'
#' * \code{lwr}: lower limit of the \code{1-alpha}
#' prediction interval on the response,
#'
#' * \code{pred_err}: prediction error calculated as \code{y-y_hat},
#'
#' * \code{pred_err_sup}: upper limit of the \code{1-alpha}
#' prediction interval on the prediction error,
#'
#' * \code{pred_err_inf}: lower limit of the \code{1-alpha}
#' prediction interval on the prediction error.
#'
#' @export
#'
#' @seealso \code{\link{control_charts_pca}}, \code{\link{regr_cc_sof}}
#'
#' @examples
#' \dontrun{
#' #' 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)
#' }
#'
control_charts_sof_pc <- function(mod,
y_test,
mfdobj_x_test,
mfdobj_x_tuning = NULL,
alpha = list(
T2 = .0125,
spe = .0125,
y = .025),
limits = "standard",
seed,
nfold = NULL,
ncores = 1) {
.Deprecated("regr_cc_sof")
regr_cc_sof(object = mod,
y_new = y_test,
mfdobj_x_new = mfdobj_x_test,
mfdobj_x_tuning = mfdobj_x_tuning,
y_tuning = NULL,
alpha = alpha,
parametric_limits = TRUE,
include_covariates = TRUE)
}
#' Plot control charts
#'
#' This function takes as input a data frame produced
#' with functions such as
#' \code{\link{control_charts_pca}} and \code{\link{control_charts_sof_pc}} and
#' produces a ggplot with the desired control charts, i.e.
#' it plots a point for each
#' observation in the phase II data set against
#' the corresponding control limits.
#'
#' @param cclist
#' A \code{data.frame} produced by
#' \code{\link{control_charts_pca}}, \code{\link{control_charts_sof_pc}}
#' \code{\link{regr_cc_fof}}, or \code{\link{regr_cc_sof}}.
#' @param nobsI
#' An integer indicating the first observations that are plotted in gray.
#' It is useful when one wants to plot the phase I data set together
#' with the phase II data. In that case, one needs to indicate the number
#' of phase I observations included in \code{cclist}.
#' Default is zero.
#'
#' @return A ggplot with the functional control charts.
#'
#' @details Out-of-control points are signaled by colouring them in red.
#'
#' @export
#' @examples
#' library(funcharts)
#' data("air")
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y <- get_mfd_list(air["NO2"],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y1 <- mfdobj_y[1:60]
#' mfdobj_y_tuning <- mfdobj_y[61:90]
#' mfdobj_y2 <- mfdobj_y[91:100]
#' mfdobj_x1 <- mfdobj_x[1:60]
#' mfdobj_x_tuning <- mfdobj_x[61:90]
#' mfdobj_x2 <- mfdobj_x[91:100]
#' mod_fof <- fof_pc(mfdobj_y1, mfdobj_x1)
#' cclist <- regr_cc_fof(mod_fof,
#' mfdobj_y_new = mfdobj_y2,
#' mfdobj_x_new = mfdobj_x2,
#' mfdobj_y_tuning = NULL,
#' mfdobj_x_tuning = NULL)
#' plot_control_charts(cclist)
#'
plot_control_charts <- function(cclist, nobsI = 0) {
if (!is.data.frame(cclist)) {
stop(paste0("cclist must be a data frame ",
"containing data to produce control charts."))
}
df_hot <- NULL
if (!is.null(cclist$T2)) {
df_hot <- data.frame(statistic = cclist$T2,
lcl = 0,
ucl = cclist$T2_lim) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "HOTELLING~T2~CONTROL~CHART")
hot_range <- df_hot %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_hot <- df_hot %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - hot_range * 0.2,
statistic > ucl ~ statistic + hot_range * 0.2,
TRUE ~ statistic,
))
}
df_spe <- NULL
if (!is.null(cclist$spe)) {
df_spe <- data.frame(statistic = cclist$spe,
lcl = 0,
ucl = cclist$spe_lim) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "SPE~CONTROL~CHART")
spe_range <- df_spe %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_spe <- df_spe %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - spe_range * 0.2,
statistic > ucl ~ statistic + spe_range * 0.2,
TRUE ~ statistic,
))
}
df_y <- NULL
if (!is.null(cclist$pred_err)) {
df_y <- data.frame(statistic = cclist$pred_err,
lcl = cclist$pred_err_inf,
ucl = cclist$pred_err_sup)
y_range <- df_y %>%
select(c(.data$statistic, .data$ucl, .data$lcl)) %>%
range() %>%
diff()
df_y <- df_y %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - y_range * 0.2,
statistic > ucl ~ statistic + y_range * 0.2,
TRUE ~ statistic,
))
df_y <- df_y %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl | .data$statistic < .data$lcl,
type = "REGRESSION~CONTROL~CHART")
}
df_hot_x <- NULL
if (!is.null(cclist$T2_x)) {
df_hot_x <- data.frame(statistic = cclist$T2_x,
lcl = 0,
ucl = cclist$T2_lim_x) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "HOTELLING~T2~CONTROL~CHART~(COVARIATES)")
hot_range <- df_hot_x %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_hot_x <- df_hot_x %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - hot_range * 0.2,
statistic > ucl ~ statistic + hot_range * 0.2,
TRUE ~ statistic,
))
}
df_spe_x <- NULL
if (!is.null(cclist$spe_x)) {
df_spe_x <- data.frame(statistic = cclist$spe_x,
lcl = 0,
ucl = cclist$spe_lim_x) %>%
mutate(id = seq_len(n()),
ooc = .data$statistic > .data$ucl,
type = "SPE~CONTROL~CHART~(COVARIATES)")
spe_range <- df_spe_x %>%
select(.data$statistic, .data$ucl, .data$lcl) %>%
range() %>%
diff()
df_spe_x <- df_spe_x %>%
mutate(ytext = case_when(
statistic < lcl ~ statistic - spe_range * 0.2,
statistic > ucl ~ statistic + spe_range * 0.2,
TRUE ~ statistic,
))
}
plot_list <- list()
if (!is.null(cclist$T2)) {
plot_list$p_hot <- ggplot(df_hot, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_hot, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(T2~statistic)) +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART))
}
if (!is.null(cclist$spe)) {
plot_list$p_spe <- ggplot(df_spe, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_spe, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(SPE~statistic)) +
ggtitle(expression(SPE~CONTROL~CHART))
}
if (!is.null(cclist$pred_err)) {
plot_list$p_y <- ggplot(df_y, aes(x = .data$id, y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_line(aes(y = .data$lcl), lty = 2) +
geom_line(aes(y = .data$ucl), lty = 2) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_y, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression("Regression residuals")) +
ggtitle(expression(REGRESSION~CONTROL~CHART))
}
if (!is.null(cclist$T2_x)) {
plot_list$p_hot_x <- ggplot(df_hot_x, aes(x = .data$id,
y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_hot_x, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(T2~statistic)) +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART~(COVARIATES)))
plot_list$p_hot <- plot_list$p_hot +
ggtitle(expression(HOTELLING~T2~CONTROL~CHART~(RESPONSE)))
}
if (!is.null(cclist$spe_x)) {
plot_list$p_spe_x <- ggplot(df_spe_x, aes(x = .data$id,
y = .data$statistic)) +
geom_line() +
geom_point(aes(colour = .data$ooc)) +
geom_blank(aes(y = 0)) +
geom_point(aes(y = .data$ucl),
pch = "-", size = 5) +
theme_bw() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
theme(legend.position = "none",
plot.title = element_text(hjust = 0.5)) +
geom_text(aes(y = .data$ytext, label = .data$id),
data = filter(df_spe_x, .data$ooc),
size = 3) +
xlab("Observation") +
ylab(expression(SPE~statistic)) +
ggtitle(expression(SPE~CONTROL~CHART~(COVARIATES)))
plot_list$p_spe <- plot_list$p_spe +
ggtitle(expression(SPE~CONTROL~CHART~(RESPONSE)))
}
p <- patchwork::wrap_plots(plot_list, ncol = 1) &
scale_color_manual(values = c("TRUE" = "red",
"FALSE" = "black",
"phase1" = "grey")) &
scale_x_continuous(
limits = c(0, nrow(cclist) + 1),
breaks = seq(1, nrow(cclist), by = round(nrow(cclist) / 50) + 1),
expand = c(0.015, 0.015))
if (nobsI > 0) {
nplots <- length(p$patches$plots) + 1
for (jj in seq_len(nplots)) {
p[[jj]]$data$ooc[seq_len(nobsI)] <- "phase1"
p[[jj]]$layers[[5]]$data <- p[[jj]]$layers[[5]]$data %>%
filter(.data$id > nobsI) %>%
as.data.frame()
p[[jj]] <- p[[jj]] +
geom_point(aes_string(y = "ucl"),
pch = "-",
size = 5,
col = "grey",
data = filter(p[[jj]]$data, .data$id <= nobsI)) +
geom_line(aes_string("id", "statistic"),
col = "grey",
data = filter(p[[jj]]$data, .data$id < nobsI))
}
p <- p &
geom_vline(aes(xintercept = nobsI + 1), lty = 2)
}
return(p)
}
#' Scalar-on-Function Regression Control Chart
#'
#' This function is deprecated. Use \code{\link{regr_cc_sof}}.
#' This function builds a data frame needed
#' to plot the scalar-on-function regression control chart,
#' based on a fitted function-on-function linear regression model and
#' proposed in Capezza et al. (2020).
#' If \code{include_covariates} is \code{TRUE},
#' it also plots the Hotelling's T2 and
#' squared prediction error control charts built on the
#' multivariate functional covariates.
#'
#' The training data have already been used to fit the model.
#' An additional 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.
#'
#' @param object
#' A list obtained as output from \code{sof_pc},
#' i.e. a fitted scalar-on-function linear regression model.
#' @param mfdobj_x_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates observations.
#' @param y_new
#' A numeric vector containing the observations of
#' the scalar response variable
#' in the phase II data set.
#' @param y_tuning
#' A numeric vector containing the observations of the scalar response
#' variable in the tuning data set.
#' If NULL, the training data, i.e. the data used to
#' fit the scalar-on-function regression model,
#' are also used as the tuning data set.
#' Default is NULL.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning set of the multivariate functional data, used to estimate the
#' control chart limits.
#' If NULL, the training data, i.e. the data used to
#' fit the scalar-on-function regression model,
#' are also used as the tuning data set.
#' Default is NULL.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability.
#' If \code{include_covariates} is \code{TRUE}, i.e.,
#' also the Hotelling's T2 and SPE control charts are built
#' on the functional covariates, then the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the three control charts equal to \code{alpha/3}.
#' In this last case,
#' if you want to set manually the Type-I error probabilities,
#' then the argument \code{alpha} must be a named list
#' with three elements, named \code{T2}, \code{spe} and \code{y},
#' respectively, each containing
#' the desired Type I error probability of
#' the corresponding control chart, where \code{y} refers to the
#' regression control chart.
#' Default value is 0.05.
#' @param parametric_limits
#' If \code{TRUE}, the limits are calculated based on the normal distribution
#' assumption on the response variable, as in Capezza et al. (2020).
#' If \code{FALSE}, the limits are calculated nonparametrically as
#' empirical quantiles of the distribution of the residuals calculated
#' on the tuning data set.
#' The default value is \code{FALSE}.
#' @param include_covariates
#' If TRUE, also functional covariates are monitored through
#' \code{control_charts_pca},.
#' If FALSE, only the scalar response, conditionally on the covariates,
#' is monitored.
#' @param absolute_error
#' A logical value that, if \code{include_covariates} is TRUE, is passed
#' to \code{\link{control_charts_pca}}.
#'
#' @return
#' A \code{data.frame} with as many rows as the
#' number of functional replications in \code{mfdobj_x_new},
#' with the following columns:
#'
#' * \code{y_hat}: the predictions of the response variable
#' corresponding to \code{mfdobj_x_new},
#'
#' * \code{y}: the same as the argument \code{y_new} given as input
#' to this function,
#'
#' * \code{lwr}: lower limit of the \code{1-alpha} prediction interval
#' on the response,
#'
#' * \code{pred_err}: prediction error calculated as \code{y-y_hat},
#'
#' * \code{pred_err_sup}: upper limit of the \code{1-alpha} prediction interval
#' on the prediction error,
#'
#' * \code{pred_err_inf}: lower limit of the \code{1-alpha} prediction interval
#' on the prediction error.
#'
#' @export
#'
#' @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.
#' \emph{Applied Stochastic Models in Business and Industry},
#' 36(3):477--500.
#' <doi:10.1002/asmb.2507>
#'
#' @examples
#' library(funcharts)
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' y <- rowMeans(air$NO2)
#' y1 <- y[1:80]
#' y2 <- y[81:100]
#' mfdobj_x1 <- mfdobj_x[1:80]
#' mfdobj_x2 <- mfdobj_x[81:100]
#' mod <- sof_pc(y1, mfdobj_x1)
#' cclist <- regr_cc_sof(object = mod,
#' y_new = y2,
#' mfdobj_x_new = mfdobj_x2)
#' plot_control_charts(cclist)
#'
regr_cc_sof <- function(object,
y_new,
mfdobj_x_new,
y_tuning = NULL,
mfdobj_x_tuning = NULL,
alpha = 0.05,
parametric_limits = FALSE,
include_covariates = FALSE,
absolute_error = FALSE) {
if (!is.list(object)) {
stop("object must be a list produced by sof_pc.")
}
if (!identical(names(object), c(
"mod",
"pca",
"beta_fd",
"residuals",
"components",
"selection",
"single_min_variance_explained",
"tot_variance_explained",
"gcv",
"PRESS"
))) {
stop("object must be a list produced by sof_pc.")
}
if (!is.numeric(y_new)) {
stop("y_new must be numeric.")
}
if (!is.null(mfdobj_x_new)) {
if (!is.mfd(mfdobj_x_new)) {
stop("mfdobj_x_new must be an object from mfd class.")
}
if (dim(mfdobj_x_new$coefs)[3] != dim(object$pca$data$coefs)[3]) {
stop(paste0("mfdobj_x_new must have the same number of variables ",
"as training data."))
}
}
nobsx <- dim(mfdobj_x_new$coefs)[2]
nobsy <- length(y_new)
if (nobsx != nobsy) {
stop(paste0("y_new and mfdobj_x_new must have ",
"the same number of observations."))
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha)) {
if (include_covariates) {
alpha <- list(T2 = alpha / 3, spe = alpha / 3, y = alpha / 3)
} else {
alpha <- list(y = alpha)
}
}
if (is.list(alpha)) {
if (include_covariates) {
if (!all(names(alpha) %in% c("T2", "spe", "y")) |
!all(c("T2", "spe", "y") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is TRUE,
it must be named with names T2, spe, y.")
}
}
if (!include_covariates) {
if (!all(names(alpha) %in% "y") |
!all("y" %in% names(alpha))) {
stop("if alpha is a list and include_covariates is FALSE,
it must be named with name y.")
}
}
}
y_hat <- predict_sof_pc(object = object,
y_new = y_new,
mfdobj_x_new = mfdobj_x_new,
alpha = alpha$y)
ret <- data.frame(
y_hat,
pred_err_sup = y_hat$upr - y_hat$fit,
pred_err_inf = y_hat$lwr - y_hat$fit)
if (!parametric_limits) {
if (is.null(y_tuning) | is.null(mfdobj_x_tuning)) {
fml <- formula(object$mod)
response_name <- all.vars(fml)[1]
y_tuning <- object$mod$model[, response_name]
mfdobj_x_tuning <- object$pca$data
}
y_hat_tuning <- predict_sof_pc(object = object,
y_new = y_tuning,
mfdobj_x_new = mfdobj_x_tuning,
alpha = alpha$y)
upr_lim <- as.numeric(quantile(y_hat_tuning$pred_err, 1 - alpha$y/2))
lwr_lim <- as.numeric(quantile(y_hat_tuning$pred_err, alpha$y/2))
ret$pred_err_sup <- upr_lim
ret$pred_err_inf <- lwr_lim
}
if (include_covariates) {
alpha_x <- alpha[c("T2", "spe")]
ccpca <- control_charts_pca(pca = object$pca,
components = object$components,
tuning_data = mfdobj_x_tuning,
newdata = mfdobj_x_new,
alpha = alpha_x,
limits = "standard",
absolute_error = absolute_error)
ret <- cbind(ccpca, ret)
} else {
ret <- cbind(data.frame(id = mfdobj_x_new$fdnames[[2]]), ret)
}
return(ret)
}
#' Functional Regression Control Chart
#'
#' It builds a data frame needed to plot the
#' Functional Regression Control Chart
#' introduced in Centofanti et al. (2021),
#' for monitoring a functional quality characteristic adjusted for
#' by the effect of multivariate functional covariates,
#' based on a fitted
#' function-on-function linear regression model.
#' The training data have already been used to fit the model.
#' An optional 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 control charts.
#' It also allows to jointly monitor the multivariate functional covariates.
#'
#' @param object
#' A list obtained as output from \code{fof_pc},
#' i.e. a fitted function-on-function linear regression model.
#' @param mfdobj_y_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional response
#' observations to be monitored.
#' @param mfdobj_x_new
#' An object of class \code{mfd} containing
#' the phase II data set of the functional covariates
#' observations to be monitored.
#' @param mfdobj_y_tuning
#' An object of class \code{mfd} containing
#' the tuning data set of the functional response observations,
#' used to estimate the control chart limits.
#' If NULL, the training data, i.e. the data used to fit the
#' function-on-function linear regression model,
#' are also used as the tuning data set, i.e.
#' \code{mfdobj_y_tuning=object$pca_y$data}.
#' Default is NULL.
#' @param mfdobj_x_tuning
#' An object of class \code{mfd} containing
#' the tuning data set of the functional covariates observations,
#' used to estimate the control chart limits.
#' If NULL, the training data, i.e. the data used to fit the
#' function-on-function linear regression model,
#' are also used as the tuning data set, i.e.
#' \code{mfdobj_x_tuning=object$pca_x$data}.
#' Default is NULL.
#' @param alpha
#' If it is a number between 0 and 1,
#' it defines the overall type-I error probability.
#' By default, it is equal to 0.05 and the Bonferroni correction
#' is applied by setting the type-I error probabilities equal to
#' \code{alpha/2} in the Hotelling's T2 and SPE control charts.
#' If \code{include_covariates} is \code{TRUE}, i.e.,
#' the Hotelling's T2 and SPE control charts are built
#' also on the multivariate functional covariates, then the Bonferroni
#' correction is applied by setting the type-I error probability
#' in the four control charts equal to \code{alpha/4}.
#' If you want to set manually the Type-I error probabilities,
#' then the argument \code{alpha} must be a named list
#' with elements named as \code{T2}, \code{spe},
#' \code{T2_x} and, \code{spe_x}, respectively, containing
#' the desired Type I error probability of
#' the T2 and SPE control charts for the functional response and
#' the multivariate functional covariates, respectively.
#' @param include_covariates
#' If TRUE, also functional covariates are monitored through
#' \code{control_charts_pca},.
#' If FALSE, only the functional response, conditionally on the covariates,
#' is monitored.
#' @param absolute_error
#' A logical value that, if \code{include_covariates} is TRUE, is passed
#' to \code{\link{control_charts_pca}}.
#'
#' @return
#' A \code{data.frame} containing the output of the
#' function \code{control_charts_pca} applied to
#' the prediction errors.
#' @export
#'
#' @seealso \code{\link{control_charts_pca}}
#'
#' @references
#' Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021)
#' Functional Regression Control Chart.
#' \emph{Technometrics}, 63(3), 281--294. <doi:10.1080/00401706.2020.1753581>
#'
#' @examples
#' library(funcharts)
#' data("air")
#' air <- lapply(air, function(x) x[1:100, , drop = FALSE])
#' fun_covariates <- c("CO", "temperature")
#' mfdobj_x <- get_mfd_list(air[fun_covariates],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y <- get_mfd_list(air["NO2"],
#' n_basis = 15,
#' lambda = 1e-2)
#' mfdobj_y1 <- mfdobj_y[1:60]
#' mfdobj_y_tuning <- mfdobj_y[61:90]
#' mfdobj_y2 <- mfdobj_y[91:100]
#' mfdobj_x1 <- mfdobj_x[1:60]
#' mfdobj_x_tuning <- mfdobj_x[61:90]
#' mfdobj_x2 <- mfdobj_x[91:100]
#' mod_fof <- fof_pc(mfdobj_y1, mfdobj_x1)
#' cclist <- regr_cc_fof(mod_fof,
#' mfdobj_y_new = mfdobj_y2,
#' mfdobj_x_new = mfdobj_x2,
#' mfdobj_y_tuning = NULL,
#' mfdobj_x_tuning = NULL)
#' plot_control_charts(cclist)
#'
regr_cc_fof <- function(object,
mfdobj_y_new,
mfdobj_x_new,
mfdobj_y_tuning = NULL,
mfdobj_x_tuning = NULL,
alpha = 0.05,
include_covariates = FALSE,
absolute_error = FALSE) {
if (!is.list(object)) {
stop("object must be a list produced by fof_pc.")
}
if (!identical(names(object), c(
"mod",
"beta_fd",
"fitted.values",
"residuals_original_scale",
"residuals",
"type_residuals",
"pca_x",
"pca_y",
"pca_res",
"components_x",
"components_y",
"components_res",
"y_standardized",
"tot_variance_explained_x",
"tot_variance_explained_y",
"tot_variance_explained_res",
"get_studentized_residuals"
))) {
stop("object must be a list produced by fof_pc.")
}
if (!(is.numeric(alpha) | is.list(alpha))) {
stop("alpha must be either a number or a list")
}
if (is.numeric(alpha) & length(alpha) > 1) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.numeric(alpha) & !all(alpha > 0 & alpha < 1)) {
stop("alpha must be a single number, between 0 and 1, or a list.")
}
if (is.list(alpha)) {
if (include_covariates) {
if (!all(names(alpha) %in% c("T2_x", "spe_x", "T2", "spe")) |
!all(c("T2_x", "spe_x", "T2", "spe") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is TRUE,
it must be named with names T2_x spe_x, T2, spe.")
}
}
if (!include_covariates) {
if (!all(names(alpha) %in% c("T2", "spe")) |
!all(c("T2", "spe") %in% names(alpha))) {
stop("if alpha is a list and include_covariates is FALSE,
it must be named with names T2 and spe.")
}
}
alpha_x <- alpha[c("T2_x", "spe_x")]
alpha_y <- alpha[c("T2", "spe")]
}
if (is.numeric(alpha)) {
alpha_y <- list(T2 = alpha / 4, spe = alpha / 4)
if (include_covariates) {
alpha_x <- list(T2_x = alpha / 4, spe_x = alpha / 4)
}
}
if (is.null(mfdobj_y_tuning) | is.null(mfdobj_x_tuning)) {
mfdobj_y_tuning <- object$pca_y$data
mfdobj_x_tuning <- object$pca_x$data
}
tuning <- predict_fof_pc(
object = object,
mfdobj_y_new = mfdobj_y_tuning,
mfdobj_x_new = mfdobj_x_tuning)
phase_II <- predict_fof_pc(
object = object,
mfdobj_y_new = mfdobj_y_new,
mfdobj_x_new = mfdobj_x_new)
out <- control_charts_pca(
pca = object$pca_res,
components = object$components_res,
tuning_data = tuning$pred_error,
newdata = phase_II$pred_error,
alpha = alpha_y,
limits = "standard"
)
ret <- out %>%
select(-contains("contribution_"))
if (include_covariates) {
alpha_x <- list(T2 = alpha_x$T2, spe = alpha_x$spe)
ret_covariates <- control_charts_pca(
pca = object$pca_x,
components = object$components_x,
tuning_data = mfdobj_x_tuning,
newdata = mfdobj_x_new,
alpha = alpha_x,
limits = "standard",
absolute_error = absolute_error
) %>%
rename("T2_x" = "T2",
"spe_x" = "spe",
"T2_lim_x" = "T2_lim",
"spe_lim_x" = "spe_lim")
ret <- cbind(ret, select(ret_covariates, -"id"))
}
return(ret)
}
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