funcharts: Functional Control Charts

Description

Provides functional control charts for statistical process monitoring of functional data, using the methods of Capezza et al. (2020) doi:10.1002/asmb.2507, Centofanti et al. (2021) doi:10.1080/00401706.2020.1753581, Capezza et al. (2024) doi:10.1080/00401706.2024.2327346, Capezza et al. (2024) doi:10.1080/00224065.2024.2383674, Centofanti et al. (2022) doi:10.48550/arXiv.2205.06256. The package is thoroughly illustrated in the paper of Capezza et al (2023) doi:10.1080/00224065.2023.2219012.

Author(s)

Maintainer: Christian Capezza christian.capezza@unina.it

Authors:

• Fabio Centofanti

• Antonio Lepore

• Biagio Palumbo

Other contributors:

• Alessandra Menafoglio [contributor]

• Simone Vantini [contributor]

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. (2021) Functional Regression Control Chart. Technometrics, 63(3):281–294. doi:10.1080/00401706.2020.1753581

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Capezza, C., Capizzi, G., Centofanti, F., Lepore, A., Palumbo, B. (2025) An Adaptive Multivariate Functional EWMA Control Chart. Journal of Quality Technology, 57(1):1–15, doi:https://doi.org/10.1080/00224065.2024.2383674.

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

See Also

Useful links:

• https://github.com/unina-sfere/funcharts

• Report bugs at https://github.com/unina-sfere/funcharts/issues

Phase I of the Adaptive Multivariate Functional Control Chart (AMFCC).

Description

This function implements the design phase (Phase I) of the Adaptive Multivariate Functional Control Chart.

Usage

AMFCC_PhaseI(
  data_tra,
  data_tun = NULL,
  grid,
  q = 30,
  par_seq_list = list(10^seq(-7, 2, l = 10), c(0.5, 0.7, 0.8, 0.9, 0.99)),
  alpha_diagn = 0.05,
  alpha_mon = 0.05,
  ncores = 1
)

Arguments

Value

A list containing the following arguments:

• statistics_IC: A matrix with the values of the Hotelling T^2-type statistics for each observation and parameter combination.

• p_values_combined: A list with two elements containing the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• CL: The control limits for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• contributions_IC: A list where each element corresponds to a variable and is a matrix with the contributions to the Hotelling T^2-type statistics for each observation and parameter combination.

• p_values_combined_cont: A list where each element corresponds to a variable and is a list of two elements containing the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• CL_cont: The control limits for the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• par_seq_list: The list of the sequences of the tuning parameters.

• q: The dimension of the set of B-spline functions.

• basis: The basis functions used for the functional data representation.

• grid: The vector of time points where the curves are sampled.

• comb_list_tot: The matrix with all the parameter combinations.

• mod_pca_list: The list of the MFPCA models for each value of lambda_s.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, and B. Palumbo (2025). An Adaptive Multivariate Functional Control Chart. Accepted for publication in Technometrics.

Examples

library(funcharts)
N <- 10
l_grid <- 10
p <- 2
grid <- seq(0, 1, l = l_grid)


Xall_tra <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  correlation_type_x = c("Bessel", "Gaussian")
)
X_tra <-
  data.frame(
    x = c(Xall_tra$X_list[[1]], Xall_tra$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

Xall_II <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  shift_type_x = list("A", "B"),
  d_x = c(10, 10),
  correlation_type_x = c("Bessel", "Gaussian")
)

X_II <-
  data.frame(
    x = c(Xall_II$X_list[[1]], Xall_II$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

# AMFCC -------------------------------------------------------------------
print("AMFCC")

mod_phaseI_AMFCC <- AMFCC_PhaseI(
  data_tra = X_tra,
  data_tun =
    NULL,
  grid = grid,
  ncores = 1
)

mod_phaseII_AMFCC <- AMFCC_PhaseII(data = X_II,
mod_Phase_I = mod_phaseI_AMFCC,
ncores = 1)

plot(mod_phaseII_AMFCC)
plot(mod_phaseII_AMFCC,type='cont',ind_obs=1)

Phase II of the Adaptive Multivariate Functional Control Chart (AMFCC).

Description

This function implements the monitoring phase (Phase II) of the Adaptive Multivariate Functional Control Chart.

Usage

AMFCC_PhaseII(data = NULL, mod_Phase_I, ncores = 1)

Arguments

      var: vector of the variable indexes.

      curve: vector of the curve indexes.

      timeindex: vector of the time indexes corresponding to given elements of \code{grid}.

      x: concatenated vector of the observed curves.

Value

A list containing the following arguments:

• ARL: The average run length (ARL) for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• ARL_cont: The average run length for the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• statistics: A matrix with the values of the Hotelling T^2-type statistics for each observation and parameter combination.

• contributions: A list where each element is a matrix with the contributions to the Hotelling T^2-type statistics for each observation and parameter combination.

• p_values_combined: A list with two elements containing the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• p_values_combined_cont: A list where each element is a list of two elements containing the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• CL: The control limits for the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

• CL_cont: The control limits for the contribution to the monitoring statistics obtained with the Fisher omnibus and Tippett combining functions.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, and B. Palumbo (2025). An Adaptive Multivariate Functional Control Chart. Accepted for publication in Technometrics.

Examples

library(funcharts)
N <- 10
l_grid <- 10
p <- 2
grid <- seq(0, 1, l = l_grid)


Xall_tra <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  correlation_type_x = c("Bessel", "Gaussian")
)
X_tra <-
  data.frame(
    x = c(Xall_tra$X_list[[1]], Xall_tra$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

Xall_II <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  shift_type_x = list("A", "B"),
  d_x = c(10, 10),
  correlation_type_x = c("Bessel", "Gaussian")
)

X_II <-
  data.frame(
    x = c(Xall_II$X_list[[1]], Xall_II$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

# AMFCC -------------------------------------------------------------------
print("AMFCC")

mod_phaseI_AMFCC <- AMFCC_PhaseI(
  data_tra = X_tra,
  data_tun =
    NULL,
  grid = grid,
  ncores = 1
)

mod_phaseII_AMFCC <- AMFCC_PhaseII(data = X_II,
mod_Phase_I = mod_phaseI_AMFCC,
ncores = 1)

plot(mod_phaseII_AMFCC)
plot(mod_phaseII_AMFCC,type='cont',ind_obs=1)

Adaptive Multivariate Functional EWMA control chart - Phase I

Description

This function performs Phase I of the Adaptive Multivariate Functional EWMA (AMFEWMA) control chart proposed by Capezza et al. (2024)

Usage

AMFEWMA_PhaseI(
  mfdobj,
  mfdobj_tuning,
  lambda = NULL,
  k = NULL,
  ARL0 = 200,
  bootstrap_pars = list(n_seq = 200, l_seq = 2000),
  optimization_pars = list(lambda_grid = c(0.1, 0.2, 0.3, 0.5, 1), k_grid = c(1, 2, 3,
    4), epsilon = 0.1, sd_small = 0.25, sd_big = 2),
  discrete_grid_length = 25,
  score_function = "huber",
  fev = 0.9,
  n_skip = 100
)

Arguments

Value

A list with the following elements. lambda is the selected lambda parameter. k is the selected k parameter. mod_1 contains the estimated Phase I model. It is a list with the following elements.

• mfdobj the mfdobj object passed as input to this function,

• mfdobj_tuning the mfdobj_tuning object passed as input to this function,

• inv_sigmaY_reg: the matrix containing the discretized version of the function K^*(s,t) defined in Equation (9) of Capezza et al. (2024),

• mean_mfdobj: the estimated mean function,

• h: the calculated upper control limit of the AMFEWMA control chart,

• ARL0: the estimated in-control ARL, which should be close to the nominal value passed as input to this function,

• lambda: the lambda parameter selected by the optimization procedure described in Section 2.4 of Capezza et al. (2024).

• k: The function C_j(t)=k sigma_j(t) appearing in the score functions (7) and (8) of Capezza et al. (2024).

• grid_points: the grid containing the points over which the functional data are discretized before computing the AMFEWMA monitoring statistic and estimating all the model parameters.

• V2_mat: the n_seqXl_seq matrix containing, in each column, the AMFEWMA monitoring statistic values of each bootstrap sequence. This matrix is used to set the control chart limit h to ensure that the desired average run length is achieved.

• n_skip: the n_skip input parameter passed to this function,

• huber: if the input parameter score_function is "huber", this is TRUE, else is FALSE,

• vectors: the discretized eigenfunctions psi_l(t) of the covariance function, appearing in Equation (9) of Capezza et al. (2024).

• values: the eigenvalues rho_l of the covariance function, appearing in Equation (9) of Capezza et al. (2024).

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Capizzi, G., Centofanti, F., Lepore, A., Palumbo, B. (2025) An Adaptive Multivariate Functional EWMA Control Chart. Journal of Quality Technology, 57(1):1–15, doi:https://doi.org/10.1080/00224065.2024.2383674.

Lucas, J. M., Saccucci, M. S. (1990) Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12.

Examples

## Not run: set.seed(0)
library(funcharts)
dat_I <- simulate_mfd(nobs = 1000,
                      correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                      sd_x = c(0.3, 0.3, 0.3))
dat_tun <- simulate_mfd(nobs = 1000,
                        correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                        sd_x = c(0.3, 0.3, 0.3))
dat_II <- simulate_mfd(nobs = 200,
                       correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                       shift_type_x = c("C", "C", "C"),
                       d_x = c(2, 2, 2),
                       sd_x = c(0.3, 0.3, 0.3))
mfdobj_I <- get_mfd_list(dat_I$X_list)
mfdobj_tun <- get_mfd_list(dat_tun$X_list)
mfdobj_II <- get_mfd_list(dat_II$X_list)

p <- plot_mfd(mfdobj_I[1:100])
lines_mfd(p, mfdobj_II, col = "red")

mod <- AMFEWMA_PhaseI(mfdobj = mfdobj_I, mfdobj_tuning = mfdobj_tun)
print(mod$k)
cc <- AMFEWMA_PhaseII(mfdobj_2 = rbind_mfd(mfdobj_I[1:100], mfdobj_II),
                      mod_1 = mod)
plot_control_charts(cc$cc, nobsI = 100)

## End(Not run)

Adaptive Multivariate Functional EWMA control chart - Phase II

Description

This function performs Phase II of the Adaptive Multivariate Functional EWMA (AMFEWMA) control chart proposed by Capezza et al. (2024)

Usage

AMFEWMA_PhaseII(mfdobj_2, mod_1, n_seq_2 = 1, l_seq_2 = 2000)

Arguments

Value

A list with the following elements.

• ARL_2: the average run length estimated over the bootstrap sequences. If n_seq_2 is 1, it is simply the run length observed over the Phase II sequence, i.e., the number of observations up to the first alarm,

• RL: the run length observed over the Phase II sequence, i.e., the number of observations up to the first alarm,

• V2: a list with length n_seq_2, containing the AMFEWMA monitoring statistic in Equation (8) of Capezza et al. (2024), calculated in each bootstrap sequence, until the first alarm.

• cc: a data frame with the information needed to plot the AMFEWMA control chart in Phase II, with the following columns. id contains the id of each multivariate functional observation, amfewma_monitoring_statistic contains the AMFEWMA monitoring statistic values calculated on the Phase II sequence, amfewma_monitoring_statistic_lim is the upper control limit.

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Capizzi, G., Centofanti, F., Lepore, A., Palumbo, B. (2025) An Adaptive Multivariate Functional EWMA Control Chart. Journal of Quality Technology, 57(1):1–15, doi:https://doi.org/10.1080/00224065.2024.2383674.

Examples

## Not run: set.seed(0)
library(funcharts)
dat_I <- simulate_mfd(nobs = 1000,
                      correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                      sd_x = c(0.3, 0.3, 0.3))
dat_tun <- simulate_mfd(nobs = 1000,
                        correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                        sd_x = c(0.3, 0.3, 0.3))
dat_II <- simulate_mfd(nobs = 200,
                       correlation_type_x = c("Bessel", "Bessel", "Bessel"),
                       shift_type_x = c("C", "C", "C"),
                       d_x = c(2, 2, 2),
                       sd_x = c(0.3, 0.3, 0.3))
mfdobj_I <- get_mfd_list(dat_I$X_list)
mfdobj_tun <- get_mfd_list(dat_tun$X_list)
mfdobj_II <- get_mfd_list(dat_II$X_list)

p <- plot_mfd(mfdobj_I[1:100])
lines_mfd(p, mfdobj_II, col = "red")

mod <- AMFEWMA_PhaseI(mfdobj = mfdobj_I, mfdobj_tuning = mfdobj_tun)
print(mod$lambda)
print(mod$k)
cc <- AMFEWMA_PhaseII(mfdobj_2 = rbind_mfd(mfdobj_I[1:100], mfdobj_II),
                      mod_1 = mod)
plot_control_charts(cc$cc, nobsI = 100)

## End(Not run)

Phase I of the FRTM method.

Description

This function implements the design phase (Phase I) of FRTM method.

Usage

FRTM_PhaseI(
  data_tra,
  data_tun = NULL,
  alpha = 0.05,
  n_basis_xall = 30,
  control.FDTW = list(),
  control.mFPCA = list(),
  control.rtr = list(),
  ncores = 1,
  print = TRUE
)

Arguments

Value

A list containing the following arguments:

T2_fd List of T^{2} functions for each observation in the tuning set.

SPE_fd List of SPE functions for each observation in the tuning set.

CL_T2 Control limit of the Hotelling's T^{2} control chart.

CL_SPE Control limit of the SPE control chart.

template_fd Template function used in the registration.

der_template_fd First derivative of the template function.

u_fd Upper extreme of the band constraint.

l_fd Lower extreme of the band constraint.

x_list_tun List, for each observation in the tuning set, of partial registered functions.

h_list_tun List, for each observation in the tuning set, of partial warping functions.

x_list List, for each observation in the training set, of partial registered functions.

h_list List, for each observation in the training set, of partial warping functions.

x_err A list containing the discrete observations for each curve of the training set.

grid_i A list of vector of time points where the curves of the training set are sampled.

x_list_smooth Smooth curves from the training set.

lambda Lambda identified through the average curve distance to obtain the OEB-FDTW solution.

par_reg Additional parameters to be used in the monitoring phase (Phase II).

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

See Also

FRTM_PhaseI

Examples

library(funcharts)
data <- simulate_data_FRTM(n_obs = 20)

data_oc <-
  simulate_data_FRTM(
    n_obs = 2,
    scenario = "1",
    shift = "OC_h",
    severity = 0.5
  )

lambda <- 10 ^ -5
max_x <- max(unlist(data$grid_i))
seq_t_tot <- seq(0, 1, length.out = 30)[-1]
seq_x <- seq(0.1, max_x, length.out = 10)


## Not run: 
  mod_phaseI_FRTM <- FRTM_PhaseI(
    data_tra =  data,
    control.FDTW = list(
      M = 30,
      N = 30,
      lambda = lambda,
      seq_t = seq_t_tot,
      iter_tem = 1,
      iter = 1
    ),
    control.rtr = list(seq_x = seq_x)
  )
  mod_phaseII_FRTM <- FRTM_PhaseII(data_oc = data_oc , mod_phaseI = mod_phaseI_FRTM)

  plot(mod_phaseI_FRTM)
  plot(mod_phaseII_FRTM)

## End(Not run)

Phase II of the FRTM method.

Description

This function implements the monitoring phase (Phase II) of FRTM method.

Usage

FRTM_PhaseII(data_oc, mod_phaseI, ncores = 1)

Arguments

Value

A list containing the following arguments:

T2_fd List of T^{2} functions for each observation.

SPE_fd List of SPE functions for each observation.

CL_T2 Control limit of the Hotelling's T^{2} control chart.

CL_SPE Control limit of the SPE control chart.

x_err A list containing the discrete observations for each curve.

grid_i A list of vector of time points where the curves are sampled.

x_list_smooth Smooth curves.

mod_phaseI An object of class mod_phaseI_FRTM obtained as output of the function FRTM_PhaseI.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

Examples

library(funcharts)
data <- simulate_data_FRTM(n_obs = 20)

data_oc <-
  simulate_data_FRTM(
    n_obs = 2,
    scenario = "1",
    shift = "OC_h",
    severity = 0.5
  )

lambda <- 10 ^ -5
max_x <- max(unlist(data$grid_i))
seq_t_tot <- seq(0, 1, length.out = 30)[-1]
seq_x <- seq(0.1, max_x, length.out = 10)


## Not run: 
  mod_phaseI_FRTM <- FRTM_PhaseI(
    data_tra =  data,
    control.FDTW = list(
      M = 30,
      N = 30,
      lambda = lambda,
      seq_t = seq_t_tot,
      iter_tem = 1,
      iter = 1
    ),
    control.rtr = list(seq_x = seq_x)
  )
  mod_phaseII_FRTM <- FRTM_PhaseII(data_oc = data_oc , mod_phaseI = mod_phaseI_FRTM)

  plot(mod_phaseI_FRTM)
  plot(mod_phaseII_FRTM)

## End(Not run)

Open-end/open-begin Functional Dynamic Time Warping (OEB-FDTW)

Description

This function implements the OEB-FDTW.

Usage

OEBFDTW(
  x_fd,
  template_fd,
  der_x_fd,
  der_template_fd,
  alpha_vec = c(0, 0.5, 1),
  fit_c = FALSE,
  N = 100,
  M = 50,
  smin = 0.01,
  smax = 1000,
  lambda = 10^-5,
  eta = 0.5,
  iter = 3,
  delta_xs = 0,
  delta_xe = 0,
  delta_ys = 0,
  delta_ye = 0,
  der_0 = NULL,
  seq_t = NULL,
  get_fd = "no",
  n_basis_x = NULL
)

Arguments

Value

A list containing the following arguments:

mod that is a list composed by

• h_fd: The estimated warping function.

• x_reg: The registered function.

• h_list: A list containing the discretized warping function for each iteration of the iterative refinement.

• min_cost: Optimal value of the objective function.

• lambda: The smoothing parameter \lambda.

• alpha: Optimal value of the parameter \alpha_i.

obj Values of the objective function for each value in alpha_vec.

fit Values of the objective function without the penalization for each value in alpha_vec.

obj_opt Optimal value of the objective function.

fit_opt Optimal value of the objective function without the penalization.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

Deriso, D. and S. Boyd (2022). A general optimization framework for dynamic time warping. Optimization and Engineering, 1-22.

Examples

set.seed(1)
data <- simulate_data_FRTM(n_obs = 100)

dom <- c(0, 1)
basis_x <- fda::create.bspline.basis(c(0, 1), nbasis = 30)
x_fd <-
  fda::smooth.basis(data$grid_i[[1]] / max(data$grid_i[[1]]), data$x_err[[1]], basis_x)$fd
template_fd <-
  fda::smooth.basis(data$grid_template, data$template, basis_x)$fd
der_x_fd <- fda::deriv.fd(x_fd, 1)
der_template_fd <- fda::deriv.fd(template_fd, 1)

mod <-
  OEBFDTW(x_fd, template_fd, der_x_fd , der_template_fd, get_fd = "x_reg")

Robust Multivariate Functional Control Charts - Phase I

Description

It performs Phase I of the Robust Multivariate Functional Control Chart (RoMFCC) as proposed by Capezza et al. (2024).

Usage

RoMFCC_PhaseI(
  mfdobj,
  mfdobj_tuning = NULL,
  functional_filter_par = list(filter = TRUE),
  imputation_par = list(method_imputation = "RoMFDI"),
  pca_par = list(fev = 0.7),
  alpha = 0.05
)

Arguments

Value

A list of the following elements that are needed in Phase II:

• T2 the Hotelling's T2 statistic values for the Phase I data set,

• SPE the SPE statistic values for the Phase I data set,

• T2_tun the Hotelling's T2 statistic values for the tuning data set,

• SPE_tun the SPE statistic values for the tuning data set,

• T2_lim the Phase II control limit of the Hotelling's T2 control chart,

• spe_lim the Phase II control limit of the SPE control chart,

• tuning TRUE if the tuning data set is provided, FALSE otherwise,

• mod_pca the final RoMFPCA model fitted on the Phase I data set,

• K = K the number of selected principal components,

• T_T2_inv if a tuning data set is provided, it returns the inverse of the covariance matrix of the first K scores, needed to calculate the Hotelling's T2 statistic for the Phase II observations.

• mean_scores_tuning_rob_mean if a tuning data set is provided, it returns the robust location estimate of the scores, needed to calculate the Hotelling's T2 and SPE statistics for the Phase II observations.

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Examples

## Not run: 
library(funcharts)
mfdobj <- get_mfd_list(air, n_basis = 5)
nobs <- dim(mfdobj$coefs)[2]
set.seed(0)
ids <- sample(1:nobs)
mfdobj1 <- mfdobj[ids[1:100]]
mfdobj_tuning <- mfdobj[ids[101:300]]
mfdobj2 <- mfdobj[ids[-(1:300)]]
mod_phase1 <- RoMFCC_PhaseI(mfdobj = mfdobj1,
                            mfdobj_tuning = mfdobj_tuning)
phase2 <- RoMFCC_PhaseII(mfdobj_new = mfdobj2,
                         mod_phase1 = mod_phase1)
plot_control_charts(phase2)

## End(Not run)

Robust Multivariate Functional Control Charts - Phase II

Description

It calculates the Hotelling's and SPE monitoring statistics needed to plot the Robust Multivariate Functional Control Chart in Phase II.

Usage

RoMFCC_PhaseII(mfdobj_new, mod_phase1)

Arguments

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 T2 statistic calculated for all observations,

• one column per each functional variable, containing its contribution to the T2 statistic,

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

• T2_lim gives the upper control limit of the Hotelling's T2 control chart,

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

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Examples

## Not run: 
library(funcharts)
mfdobj <- get_mfd_list(air, n_basis = 5)
nobs <- dim(mfdobj$coefs)[2]
set.seed(0)
ids <- sample(1:nobs)
mfdobj1 <- mfdobj[ids[1:100]]
mfdobj_tuning <- mfdobj[ids[101:300]]
mfdobj2 <- mfdobj[ids[-(1:300)]]
mod_phase1 <- RoMFCC_PhaseI(mfdobj = mfdobj1,
                            mfdobj_tuning = mfdobj_tuning)
phase2 <- RoMFCC_PhaseII(mfdobj_new = mfdobj2,
                         mod_phase1 = mod_phase1)
plot_control_charts(phase2)

## End(Not run)

Robust Multivariate Functional Data Imputation (RoMFDI)

Description

It performs Robust Multivariate Functional Data Imputation (RoMFDI) as in Capezza et al. (2024).

Usage

RoMFDI(
  mfdobj,
  method_pca = "ROBPCA",
  fev = 0.999,
  n_dataset = 3,
  update = TRUE,
  niter_update = 10,
  alpha = 0.8
)

Arguments

Value

A list with n_dataset elements. Each element is an mfd object containing mfdobj with stochastic imputation of the missing components.

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Van Ginkel, J. R., Van der Ark, L. A., Sijtsma, K., and Vermunt, J. K. (2007). Two-way imputation: a Bayesian method for estimating missing scores in tests and questionnaires, and an accurate approximation. Computational Statistics & Data Analysis, 51(8):4013–-4027.

Examples

## Not run: 
library(funcharts)
mfdobj <- get_mfd_list(air, grid = 1:24, n_basis = 13, lambda = 1e-2)
out <- functional_filter(mfdobj)
mfdobj_imp <- RoMFDI(out$mfdobj)

## End(Not run)

Extract observations and/or variables from mfd objects.

Description

Extract observations and/or variables from mfd objects.

Usage

## S3 method for class 'mfd'
mfdobj[i = TRUE, j = TRUE]

Arguments

Details

This function adapts the fda::"[.fd" function to be more robust and suitable for the mfd class. In fact, whatever the number of observations or variables you want to extract, it always returns a mfd object with a three-dimensional coef array. In other words, it behaves as you would always use the argument drop=FALSE. Moreover, you can extract observations and variables both by index numbers and by names, as you would normally do when using `[` with standard vector/matrices.

Value

a mfd object with selected observations and variables.

Examples

library(funcharts)
library(fda)

# In the following, we extract the first one/two observations/variables
# to see the difference with `[.fd`.
mfdobj <- data_sim_mfd()
fdobj <- fd(mfdobj$coefs, mfdobj$basis, mfdobj$fdnames)

# The argument `coef` in `fd`
# objects is converted to a matrix when possible.
dim(fdobj[1, 1]$coef)
# Not clear what is the second dimension:
# the number of replications or the number of variables?
dim(fdobj[1, 1:2]$coef)
dim(fdobj[1:2, 1]$coef)

# The argument `coef` in `mfd` objects is always a three-dimensional array.
dim(mfdobj[1, 1]$coef)
dim(mfdobj[1, 1:2]$coef)
dim(mfdobj[1:2, 1]$coef)

# Actually, `[.mfd` works as `[.fd` when passing also `drop = FALSE`
dim(fdobj[1, 1, drop = FALSE]$coef)
dim(fdobj[1, 1:2, drop = FALSE]$coef)
dim(fdobj[1:2, 1, drop = FALSE]$coef)

Air quality data

Description

This data set has been included from the R package FRegSigCom. The original .RData file is available at https://github.com/cran/FRegSigCom/blob/master/data/air.RData.

Data collected hourly in 355 days (days with missing values removed) in a significantly polluted area within an Italian city.

Usage

data("air")

Format

A list of 7 matrices with 355 rows and 24 columns:

NO2

Hourly observation of concentration level of NO2 in 355 days

CO

Hourly observation of concentration level of CO in 355 days

NMHC

Hourly observation of concentration level of NMHC in 355 days

NOx

Hourly observation of concentration level of NOx in 355 days

C6H6

Hourly observation of concentration level of C6H6 in 355 days

temperature

Hourly observation of concentration level of temperature in 355 days

humidity

Hourly observation of concentration level of humidity in 355 days

Source

https://archive.ics.uci.edu/ml/datasets/Air+quality

References

De Vito, S., Massera E., Piga M., Martinotto L. and Di Francia G. (2008). On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario Sensors and Actuators B: Chemical, 129: 50-757. doi:10.1016/j.snb.2007.09.060

Xin Qi and Ruiyan Luo (2019). Nonlinear function on function additive model with multiple predictor curves. Statistica Sinica, 29:719-739. doi:10.5705/ss.202017.0249

Bind variables of two Multivariate Functional Data Objects

Description

Bind variables of two Multivariate Functional Data Objects

Usage

cbind_mfd(mfdobj1, mfdobj2)

Arguments

Value

An object of class mfd, whose replications are the same of mfdobj1 and mfdobj2 and whose functional variables are the union of the functional variables in mfdobj1 and mfdobj2.

Examples

library(funcharts)
mfdobj1 <- data_sim_mfd(nvar = 3)
mfdobj2 <- data_sim_mfd(nvar = 2)
dimnames(mfdobj2$coefs)[[3]] <- mfdobj2$fdnames[[3]] <- c("var10", "var11")

plot_mfd(mfdobj1)
plot_mfd(mfdobj2)
mfdobj_cbind <- cbind_mfd(mfdobj1, mfdobj2)
plot_mfd(mfdobj_cbind)

Produce contribution plots

Description

This function produces a contribution plot from functional control charts for a given observation of a phase II data set, using ggplot.

Usage

cont_plot(cclist, id_num, which_plot = c("T2", "spe"), print_id = FALSE)

Arguments

Value

A ggplot containing the contributions of functional variables to the monitoring statistics. Each plot is a bar plot, with bars corresponding to contribution values and horizontal black segments denoting corresponding (empirical) upper limits. Bars are coloured by red if contributions exceed their limit.

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]
y_tuning <- y[61:90]
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 <- regr_cc_sof(object = mod,
                      y_new = y2,
                      mfdobj_x_new = mfdobj_x2,
                      y_tuning = y_tuning,
                      mfdobj_x_tuning = mfdobj_x_tuning,
                      include_covariates = TRUE)
get_ooc(cclist)
cont_plot(cclist, 3)

T2 and SPE control charts for multivariate functional data

Description

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.

Usage

control_charts_pca(
  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
)

Arguments

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 T2 statistic calculated for all observations,

• one column per each functional variable, containing its contribution to the T2 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 T2 control chart,

• one contribution_T2_*_lim column per each functional variable giving the limits of the contribution of that variable to the Hotelling's T2 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

Capizzi, G., & Masarotto, G. (2018). Phase I distribution-free analysis with the R package dfphase1. In Frontiers in Statistical Quality Control 12 (pp. 3-19). Springer International Publishing.

See Also

regr_cc_fof

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)

Real-time T2 and SPE control charts for multivariate functional data

Description

This function produces a list of data frames, each of them is produced by control_charts_pca and is needed to plot control charts for monitoring multivariate functional covariates each evolving up to an intermediate domain point.

Usage

control_charts_pca_mfd_real_time(
  pca_list,
  components_list = NULL,
  mfdobj_x_test,
  mfdobj_x_tuning = NULL,
  alpha = 0.05,
  limits = "standard",
  seed,
  nfold = NULL,
  tot_variance_explained = 0.9,
  single_min_variance_explained = 0,
  absolute_error = FALSE,
  ncores = 1
)

Arguments

Value

A list of data.frames each produced by control_charts_pca, corresponding to a given instant.

See Also

pca_mfd_real_time, control_charts_pca

Examples

library(funcharts)
data("air")
air1 <- lapply(air, function(x) x[1:8, , drop = FALSE])
air2 <- lapply(air, function(x) x[9:10, , drop = FALSE])
mfdobj_x1_list <- get_mfd_list_real_time(air1[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_x2_list <- get_mfd_list_real_time(air2[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
pca_list <- pca_mfd_real_time(mfdobj_x1_list)

cclist <- control_charts_pca_mfd_real_time(
  pca_list = pca_list,
  components_list = 1:3,
  mfdobj_x_test = mfdobj_x2_list)
plot_control_charts_real_time(cclist, 1)

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 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. 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.

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,
  nfold = NULL,
  ncores = 1
)

Arguments

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 T2 statistic calculated for all observations,

• one column per each functional variable, containing its contribution to the T2 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 T2 control chart,

• one contribution_T2_*_lim column per each functional variable giving the limits of the contribution of that variable to the Hotelling's T2 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.

Author(s)

C. Capezza

See Also

control_charts_pca, regr_cc_sof

Examples

## Not run: 
#' 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)

## End(Not run)

Real-time scalar-on-function regression control charts

Description

This function is deprecated. Use regr_cc_sof_real_time. This function produces a list of data frames, each of them is produced by control_charts_sof_pc and is needed to plot control charts for monitoring in real time a scalar quality characteristic adjusted for by the effect of multivariate functional covariates.

Usage

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

Arguments

Value

A list of data.frames each produced by control_charts_sof_pc, corresponding to a given instant.

See Also

sof_pc_real_time, control_charts_sof_pc

Examples

## Not run: 
library(funcharts)
data("air")
air1 <- lapply(air, function(x) x[1:8, , drop = FALSE])
air2 <- lapply(air, function(x) x[9:10, , drop = FALSE])
mfdobj_x1_list <- get_mfd_list_real_time(air1[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_x2_list <- get_mfd_list_real_time(air2[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
y1 <- rowMeans(air1$NO2)
y2 <- rowMeans(air2$NO2)
mod_list <- sof_pc_real_time(y1, mfdobj_x1_list)
cclist <- control_charts_sof_pc_real_time(
  mod_list = mod_list,
  y_test = y2,
  mfdobj_x_test = mfdobj_x2_list)
plot_control_charts_real_time(cclist, 1)

## End(Not run)

Correlation Function for Multivariate Functional Data

Description

Computes the correlation function for two multivariate functional data objects of class mfd.

Usage

cor_mfd(mfdobj1, mfdobj2 = mfdobj1)

Arguments

Details

The function calculates the correlation between all pairs of dimensions from the two multivariate functional data objects. The data is first scaled using scale_mfd, and the correlation is then computed as the covariance of the scaled data using cov_mfd.

Value

A bifd object representing the correlation function of the two input objects. The output is a collection of p^2 functional surfaces, each corresponding to the correlation between two components of the multivariate functional data.

Examples

## Not run: 
library(funcharts)
data("air")
x <- get_mfd_list(air[1:3])
cor_result <- cor_mfd(x)
plot_bifd(cor_result)

## End(Not run)

Covariance Function for Multivariate Functional Data

Description

Computes the covariance function for two multivariate functional data objects of class mfd.

Usage

cov_mfd(mfdobj1, mfdobj2 = mfdobj1)

Arguments

Details

The function calculates the covariance between all pairs of dimensions from the two multivariate functional data objects. Each covariance is represented as a functional surface in the resulting bifd object. The covariance function is useful for analyzing relationships between functional variables.

Value

A bifd object representing the covariance function of the two input objects. The output is a collection of p^2 functional surfaces, each corresponding to the covariance between two components of the multivariate functional data.

Examples

## Not run: 
library(funcharts)
data("air")
x <- get_mfd_list(air[1:3])
cov_result <- cov_mfd(x)
plot_bifd(cov_result)

## End(Not run)

Simulate multivariate functional data

Description

Simulate random coefficients and create a multivariate functional data object of class mfd. It is mainly for internal use, to check that the package functions work.

Usage

data_sim_mfd(nobs = 5, nbasis = 5, nvar = 2, seed)

Arguments

Value

A simulated object of class mfd.

Examples

library(funcharts)
data_sim_mfd()

Function-on-function linear regression based on principal components

Description

Function-on-function linear regression based on principal components. This function performs multivariate functional principal component analysis (MFPCA) to extract multivariate functional principal components from the multivariate functional covariates as well as from the functional response, then it builds a linear regression model of the response scores on the covariate scores. Both functional covariates and response are standardized before the regression. See Centofanti et al. (2021) for additional details.

Usage

fof_pc(
  mfdobj_y,
  mfdobj_x,
  tot_variance_explained_x = 0.95,
  tot_variance_explained_y = 0.95,
  tot_variance_explained_res = 0.95,
  components_x = NULL,
  components_y = NULL,
  type_residuals = "standard"
)

Arguments

Value

A list containing the following arguments:

• mod: an object of class lm that is a linear regression model where the response variables are the MFPCA scores of the response variable and the covariates are the MFPCA scores of the functional covariates. mod$coefficients contains the matrix of coefficients of the functional regression basis functions,

• beta_fd: a bifd object containing the bivariate functional regression coefficients \beta(s,t) estimated with the function-on-function linear regression model,

• fitted.values: a multivariate functional data object of class mfd with the fitted values of the functional response observations based on the function-on-function linear regression model,

• residuals_original_scale: a multivariate functional data object of class mfd with the functional residuals of the function-on-function linear regression model on the original scale, i.e. they are the difference between mfdobj_y and fitted.values,

• residuals: a multivariate functional data object of class mfd with the functional residuals of the function-on-function linear regression model, standardized or studentized depending on the argument type_residuals,

• type_residuals: the same as the provided argument,

• pca_x: an object of class pca_mfd obtained by doing MFPCA on the functional covariates,

• pca_y: an object of class pca_mfd obtained by doing MFPCA on the functional response,

• pca_res: an object of class pca_mfd obtained by doing MFPCA on the functional residuals,

• components_x: a vector of integers with the components selected in the pca_x model,

• components_y: a vector of integers with the components selected in the pca_y model,

• components_res: a vector of integers with the components selected in the pca_res model,

• y_standardized: the standardized functional response obtained doing scale_mfd(mfdobj_y),

• tot_variance_explained_x: the same as the provided argument

• tot_variance_explained_y: the same as the provided argument

• tot_variance_explained_res: the same as the provided argument

• get_studentized_residuals: a function that allows to calculate studentized residuals on new data, given the estimated function-on-function linear regression model.

Author(s)

C. Capezza, F. Centofanti

References

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3), 281–294. doi:10.1080/00401706.2020.1753581

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj <- get_mfd_list(air, lambda = 1e-2)
mfdobj_y <- mfdobj[, "NO2"]
mfdobj_x <- mfdobj[, fun_covariates]
mod <- fof_pc(mfdobj_y, mfdobj_x)

Get a list of function-on-function linear regression models estimated on functional data each evolving up to an intermediate domain point.

Description

This function produces a list of objects, each of them contains the result of applying fof_pc to a functional response variable and multivariate functional covariates evolved up to an intermediate domain point.

Usage

fof_pc_real_time(
  mfdobj_y_list,
  mfdobj_x_list,
  tot_variance_explained_x = 0.95,
  tot_variance_explained_y = 0.95,
  tot_variance_explained_res = 0.95,
  components_x = NULL,
  components_y = NULL,
  type_residuals = "standard",
  ncores = 1
)

Arguments

Value

A list of lists each produced by fof_pc, corresponding to a given instant.

Author(s)

C. Capezza, F. Centofanti

See Also

fof_pc, get_mfd_df_real_time, get_mfd_list_real_time

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
mfdobj_y_list <- get_mfd_list_real_time(air["NO2"],
                                        n_basis = 15,
                                        lambda = 1e-2,
                                        k_seq = c(0.5, 0.75, 1))
mfdobj_x_list <- get_mfd_list_real_time(air[c("CO", "temperature")],
                                        n_basis = 15,
                                        lambda = 1e-2,
                                        k_seq = c(0.5, 0.75, 1))
mod_list <- fof_pc_real_time(mfdobj_y_list, mfdobj_x_list)

Finds functional componentwise outliers

Description

It finds functional componentwise outliers as described in Capezza et al. (2024).

Usage

functional_filter(
  mfdobj,
  method_pca = "ROBPCA",
  alpha = 0.95,
  fev = 0.999,
  delta = 0.1,
  alpha_binom = 0.99,
  bivariate = TRUE,
  max_proportion_componentwise = 0.5
)

Arguments

Value

A list with two elements. The first element is an mfd object containing the original observation in the mfdobj input, but where the basis coefficients of the components identified as functional componentwise outliers are replaced by NA. The second element of the list is a list of numbers, with length equal to the number of functional variables in mfdobj. Each element of this list contains the observations of the flagged functional componentwise outliers for the corresponding functional variable.

Author(s)

C. Capezza, F. Centofanti

References

Agostinelli, C., Leung, A., Yohai, V. J., and Zamar, R. H. (2015). Robust estimation of multivariate location and scatter in the presence of componentwise and casewise contamination. Test, 24(3):441–461.

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Leung, A., Yohai, V., and Zamar, R. (2017). Multivariate location and scatter matrix estimation under componentwise and casewise contamination. Computational Statistics & Data Analysis, 111:59–76.

Examples

## Not run: 
library(funcharts)
mfdobj <- get_mfd_list(air, grid = 1:24, n_basis = 13, lambda = 1e-2)
plot_mfd(mfdobj)
out <- functional_filter(mfdobj)

## End(Not run)

Get Multivariate Functional Data from a three-dimensional array

Description

Get Multivariate Functional Data from a three-dimensional array

Usage

get_mfd_array(
  data_array,
  grid = NULL,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  ncores = 1
)

Arguments

Value

An object of class mfd. See also ?mfd for additional details on the multivariate functional data class.

See Also

get_mfd_list, get_mfd_df

Examples

library(funcharts)
library(fda)
data("CanadianWeather")
mfdobj <- get_mfd_array(CanadianWeather$dailyAv[, 1:10, ],
                        lambda = 1e-5)
plot_mfd(mfdobj)

Get a list of functional data objects each evolving up to an intermediate domain point.

Description

This function produces a list functional data objects, each evolving up to an intermediate domain point, that can be used to estimate models that allow real-time predictions of incomplete functions, from the current functional domain up to the end of the observation, and to build control charts for real-time monitoring.

It calls the function get_mfd_array for each domain point.

Usage

get_mfd_array_real_time(
  data_array,
  grid = NULL,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  k_seq = seq(from = 0.25, to = 1, length.out = 10),
  ncores = 1
)

Arguments

Value

A list of mfd objects as produced by get_mfd_array.

See Also

get_mfd_array

Examples

library(funcharts)
library(fda)
data("CanadianWeather")
fdobj <- get_mfd_array_real_time(CanadianWeather$dailyAv[, 1:5, 1:2],
                                 lambda = 1e-2)

Get Multivariate Functional Data from a data frame

Description

Get Multivariate Functional Data from a data frame

Usage

get_mfd_df(
  dt,
  domain,
  arg,
  id,
  variables,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  ncores = 1
)

Arguments

Details

Basis functions are created with fda::create.bspline.basis(domain, n_basis), i.e. B-spline basis functions of order 4 with equally spaced knots are used to create mfd objects.

The smoothing penalty lambda is provided as fda::fdPar(bs, 2, lambda), where bs is the basis object and 2 indicates that the integrated squared second derivative is penalized.

Rather than having a data frame with long format, i.e. with all functional observations in a single column for each functional variable, if all functional observations are observed on a common equally spaced grid, discrete data may be available in matrix form for each functional variable. In this case, see get_mfd_list.

Value

An object of class mfd. See also ?mfd for additional details on the multivariate functional data class.

See Also

get_mfd_list

Examples

library(funcharts)

x <- seq(1, 10, length = 25)
y11 <- cos(x)
y21 <- cos(2 * x)
y12 <- sin(x)
y22 <- sin(2 * x)
df <- data.frame(id = factor(rep(1:2, each = length(x))),
                 x = rep(x, times = 2),
                 y1 = c(y11, y21),
                 y2 = c(y12, y22))

mfdobj <- get_mfd_df(dt = df,
                     domain = c(1, 10),
                     arg = "x",
                     id = "id",
                     variables = c("y1", "y2"),
                     lambda = 1e-5)

Get a list of functional data objects each evolving up to an intermediate domain point.

Description

This function produces a list functional data objects, each evolving up to an intermediate domain point, that can be used to estimate models that allow real-time predictions of incomplete functions, from the current functional domain up to the end of the observation, and to build control charts for real-time monitoring.

It calls the function get_mfd_df for each domain point.

Usage

get_mfd_df_real_time(
  dt,
  domain,
  arg,
  id,
  variables,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  k_seq = seq(from = 0.25, to = 1, length.out = 10),
  ncores = 1
)

Arguments

Value

A list of mfd objects as produced by get_mfd_df, corresponding to a given instant.

See Also

get_mfd_df

Examples

library(funcharts)

x <- seq(1, 10, length = 25)
y11 <- cos(x)
y21 <- cos(2 * x)
y12 <- sin(x)
y22 <- sin(2 * x)
df <- data.frame(id = factor(rep(1:2, each = length(x))),
                 x = rep(x, times = 2),
                 y1 = c(y11, y21),
                 y2 = c(y12, y22))

mfdobj_list <- get_mfd_df_real_time(dt = df,
                                    domain = c(1, 10),
                                    arg = "x",
                                    id = "id",
                                    variables = c("y1", "y2"),
                                    lambda = 1e-2)

Convert a fd object into a Multivariate Functional Data object.

Description

Convert a fd object into a Multivariate Functional Data object.

Usage

get_mfd_fd(fdobj)

Arguments

Value

An object of class mfd. See also ?mfd for additional details on the multivariate functional data class.

See Also

mfd

Examples

library(funcharts)
library(fda)
bs <- create.bspline.basis(nbasis = 10)
fdobj <- fd(coef = 1:10, basisobj = bs)
mfdobj <- get_mfd_fd(fdobj)

Get Multivariate Functional Data from a list of matrices

Description

Get Multivariate Functional Data from a list of matrices

Usage

get_mfd_list(
  data_list,
  grid = NULL,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  ncores = 1
)

Arguments

Details

Basis functions are created with fda::create.bspline.basis(domain, n_basis), i.e. B-spline basis functions of order 4 with equally spaced knots are used to create mfd objects.

The smoothing penalty lambda is provided as fda::fdPar(bs, 2, lambda), where bs is the basis object and 2 indicates that the integrated squared second derivative is penalized.

Rather than having a list of matrices, you may have a data frame with long format, i.e. with all functional observations in a single column for each functional variable. In this case, see get_mfd_df.

Value

An object of class mfd. See also mfd for additional details on the multivariate functional data class.

See Also

mfd, get_mfd_list, get_mfd_array

Examples

library(funcharts)
data("air")
# Only take first 5 multivariate functional observations
# and only two variables from air
air_small <- lapply(air[c("NO2", "CO")], function(x) x[1:5, ])
mfdobj <- get_mfd_list(data_list = air_small)

Get a list of functional data objects each evolving up to an intermediate domain point.

Description

This function produces a list functional data objects, each evolving up to an intermediate domain point, that can be used to estimate models that allow real-time predictions of incomplete functions, from the current functional domain up to the end of the observation, and to build control charts for real-time monitoring.

It calls the function get_mfd_list for each domain point.

Usage

get_mfd_list_real_time(
  data_list,
  grid = NULL,
  n_basis = 30,
  n_order = 4,
  basisobj = NULL,
  Lfdobj = 2,
  lambda = NULL,
  lambda_grid = 10^seq(-10, 1, length.out = 10),
  k_seq = seq(from = 0.2, to = 1, by = 0.1),
  ncores = 1
)

Arguments

Value

A list of mfd objects as produced by get_mfd_list.

See Also

get_mfd_list

Examples

library(funcharts)
data("air")
# Only take first 5 multivariate functional observations from air
air_small <- lapply(air, function(x) x[1:5, ])
# Consider only 3 domain points: 0.5, 0.75, 1
mfdobj <- get_mfd_list_real_time(data_list = air_small,
                                 lambda = 1e-2,
                                 k_seq = c(0.5, 0.75, 1))

Get out of control observations from control charts

Description

Get out of control observations from control charts

Usage

get_ooc(cclist)

Arguments

Value

A data.frame with the same number of rows as cclist, and the same number of columns apart from the columns indicating control chart limits. Each value is TRUE if the corresponding observation is in control and FALSE otherwise.

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]
y_tuning <- y[61:90]
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 <- regr_cc_sof(object = mod,
                      y_new = y2,
                      mfdobj_x_new = mfdobj_x2,
                      y_tuning = y_tuning,
                      mfdobj_x_tuning = mfdobj_x_tuning,
                      include_covariates = TRUE)
get_ooc(cclist)

Get outliers from multivariate functional data

Description

Get outliers from multivariate functional data using the functional boxplot with the modified band depth of Sun et al. (2011, 2012). This function relies on the fbplot function of the roahd package.

Usage

get_outliers_mfd(mfdobj)

Arguments

Value

A numeric vector with the indexes of the functional observations signaled as outliers.

References

• Sun, Y., & Genton, M. G. (2011). Functional boxplots. Journal of Computational and Graphical Statistics, 20(2), 316-334.

• Sun, Y., & Genton, M. G. (2012). Adjusted functional boxplots for spatio-temporal data visualization and outlier detection. Environmetrics, 23(1), 54-64.

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:20, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
get_outliers_mfd(mfdobj_x)

Get possible outliers of a training data set of a scalar-on-function regression model.

Description

Get possible outliers of a training data set of a scalar-on-function regression model. It sets the training data set also as tuning data set for the calculation of control chart limits, and as phase II data set to compare monitoring statistics against the limits and identify possible outliers. This is only an empirical approach. It is advised to use methods appropriately designed for phase I monitoring to identify outliers.

Usage

get_sof_pc_outliers(y, mfdobj)

Arguments

Value

A character vector with the ids of functional observations signaled as possibly anomalous.

Examples

## Not run: 
library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
y <- rowMeans(air$NO2)
get_sof_pc_outliers(y, mfdobj_x)

## End(Not run)

Inner products of functional data contained in mfd objects.

Description

Inner products of functional data contained in mfd objects.

Usage

inprod_mfd(mfdobj1, mfdobj2 = NULL)

Arguments

Details

Note that L^2 inner products are not calculated for couples of functional data from different functional variables. This function is needed to calculate the inner product in the product Hilbert space in the case of multivariate functional data, which for each observation is the sum of the L^2 inner products obtained for each functional variable.

Value

a three-dimensional array of L^2 inner products. The first dimension is the number of functions in argument mfdobj1, the second dimension is the same thing for argument mfdobj2, the third dimension is the number of functional variables. If you sum values over the third dimension, you get a matrix of inner products in the product Hilbert space of multivariate functional data.

Examples

library(funcharts)
set.seed(123)
mfdobj1 <- data_sim_mfd()
mfdobj2 <- data_sim_mfd()
inprod_mfd(mfdobj1)
inprod_mfd(mfdobj1, mfdobj2)

Inner product of two multivariate functional data objects, for each observation

Description

Inner product of two multivariate functional data objects, for each observation

Usage

inprod_mfd_diag(mfdobj1, mfdobj2 = NULL)

Arguments

Value

It calculates the inner product of two multivariate functional data objects. The main function inprod of the package fda calculates inner products among all possible couples of observations. This means that, if mfdobj1 has n1 observations and mfdobj2 has n2 observations, then for each variable n1 X n2 inner products are calculated. However, often one is interested only in calculating the n inner products between the n observations of mfdobj1 and the corresponding n observations of mfdobj2. This function provides this "diagonal" inner products only, saving a lot of computation with respect to using fda::inprod and then extracting the diagonal elements. Note that the code of this function calls a modified version of fda::inprod().

Examples

mfdobj <- data_sim_mfd()
inprod_mfd_diag(mfdobj)

Confirm Object has Class mfd

Description

Check that an argument is a multivariate functional data object of class mfd.

Usage

is.mfd(mfdobj)

Arguments

Value

a logical value: TRUE if the class is correct, FALSE otherwise.

Add the plot of a new multivariate functional data object to an existing plot.

Description

Add the plot of a new multivariate functional data object to an existing plot.

Usage

lines_mfd(
  plot_mfd_obj,
  mfdobj_new,
  mapping = NULL,
  data = NULL,
  stat = "identity",
  position = "identity",
  na.rm = TRUE,
  orientation = NA,
  show.legend = NA,
  inherit.aes = TRUE,
  type_mfd = "mfd",
  y_lim_equal = FALSE,
  ...
)

Arguments

Value

A plot of the multivariate functional data object added to the existing one.

Examples

library(funcharts)
library(ggplot2)
mfdobj1 <- data_sim_mfd()
mfdobj2 <- data_sim_mfd()
p <- plot_mfd(mfdobj1)
lines_mfd(p, mfdobj_new = mfdobj2)

Mixed Functional Principal Component Analysis (mFPCA)

Description

This function implements the mFPCA.

Usage

mFPCA(x_fd, h_fd = NULL, k_weights = "equal", ncom = "ptv", par_ncom = 0.9)

Arguments

Value

A list containing the following arguments:

eigfun_fd A List of functions corresponding to the functional part of the principal components.

eigvect_sc A matrix corresponding to the scalar part of the principal components.

scores Scores corresponding to x_fd and h_fd.

values Eigenvalues corresponding to the selected principal components.

varprop Variance proportion explained by each principal component.

k_weights The vector of the four constants in the inner product computation.

x_fd_list A List of two elements: the list of the registered functions and the list of the centered log-ratio transformation of the first derivatives of the normalized warping functions.

sc_mat Two column matrix corresponding to the scalar part of the observations used.

mean_fd_list Mean functions of the functional part.

mean_sc_mat Means of the scalar part.

sd_fd_list The standard deviation of the functional part.

sd_sc_mat Standard deviations of the scalar part.

h_fd An object of class fd corresponding to the warping functions.

x_fd An object of class fd corresponding to the registered functions. ind_var Additional parameter used in FRTM_PhaseI.

Author(s)

F. Centofanti

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

Examples

library(funcharts)

data <- simulate_data_FRTM(n_obs = 100)
X <- sapply(1:100, function(ii)
  data$x_true[[ii]])
x_fd <-
  fda::smooth.basis(y = X,
                    argvals =  data$grid,
                    fda::create.bspline.basis(c(0, 1), 30))$fd
H <- sapply(1:100, function(ii)
  data$h[[ii]])
h_fd <-
  fda::smooth.basis(y = H,
                    argvals =  data$grid,
                    fda::create.bspline.basis(c(0, 1), 30))$fd
mod_mFPCA <- mFPCA(x_fd, h_fd, ncom = "ptv", par_ncom = 0.95)
plot(mod_mFPCA)

Mean Function for Multivariate Functional Data

Description

Computes the mean function for a multivariate functional data object of class mfd.

Usage

mean_mfd(mfdobj)

Arguments

Details

This function calculates the mean function for each dimension of the multivariate functional data by averaging the coefficients across all observations. The resulting object is a single observation containing the mean function for each dimension.

Value

An mfd object representing the mean function of the input data. The output contains one observation, representing the mean function for each dimension of the multivariate functional variable.

Examples

## Not run: 
library(funcharts)
data(air)
mfdobj <- get_mfd_list(air)
mean_result <- mean_mfd(mfdobj)
plot_mfd(mean_result)

## End(Not run)

Define a Multivariate Functional Data Object

Description

This is the constructor function for objects of the mfd class. It is a wrapper to fda::fd, but it forces the coef argument to be a three-dimensional array of coefficients even if the functional data is univariate. Moreover, it allows to include the original raw data from which you get the smooth functional data. Finally, it also includes the matrix of precomputed inner products of the basis functions, which can be useful to speed up computations when calculating inner products between functional observations

Usage

mfd(coef, basisobj, fdnames = NULL, raw = NULL, id_var = NULL, B = NULL)

Arguments

Details

To check that an object is of this class, use function is.mfd.

Value

A multivariate functional data object (i.e., having class mfd), which is a list with components named coefs, basis, and fdnames, as for class fd, with possibly in addition the components raw and id_var.

References

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

Examples

library(funcharts)
library(fda)
set.seed(0)
nobs <- 5
nbasis <- 10
nvar <- 2
coef <- array(rnorm(nobs * nbasis * nvar), dim = c(nbasis, nobs, nvar))
bs <- create.bspline.basis(rangeval = c(0, 1), nbasis = nbasis)
mfdobj <- mfd(coef = coef, basisobj = bs)
plot_mfd(mfdobj)

Norm of Multivariate Functional Data

Description

Norm of multivariate functional data contained in a mfd object.

Usage

norm.mfd(mfdobj)

Arguments

Value

A vector of length equal to the number of replications in mfdobj, containing the norm of each multivariate functional observation in the product Hilbert space, i.e. the sum of L^2 norms for each functional variable.

Examples

library(funcharts)
mfdobj <- data_sim_mfd()
norm.mfd(mfdobj)

Setting open-end/open-begin functional dynamic time warping (OEB-FDTW) defaults

Description

This is an internal function of package FRTM which allows controlling the parameters to implement the OEB-FDTW in the FRTM method.

Usage

par.FDTW(
  N = 100,
  M = 50,
  smin = NULL,
  smax = NULL,
  alpha_vec = c(0, 0.5, 1),
  frac_oeob = 0.1,
  eta = 0.5,
  iter = 3,
  template = "Procrustes",
  grid_tem = NULL,
  index_tem = NULL,
  iter_tem = 2,
  lambda = c(0, 10^seq(-8, -2, by = 0.25), 10^5),
  threshold = 0.01,
  seq_t = seq(0.01, 1, length.out = 100)
)

Arguments

Author(s)

F. Centofanti

References

Deriso, D. and S. Boyd (2022). A general optimization framework for dynamic time warping. Optimization and Engineering, 1-22.

See Also

FRTM_PhaseI

Examples

library(funcharts)
par.FDTW()

Setting mixed functional principal component analysis (mFPCA) defaults

Description

This is an internal function of package FRTM which allows controlling the parameters used in the mFPCA in the FRTM method.

Usage

par.mFPCA(perc_pca = 0.9, perc_basis_x_pca = 1, perc_basis_h = 0.2)

Arguments

See Also

FRTM_PhaseI

Examples

library(funcharts)
par.mFPCA()

Setting real-time registration step defaults

Description

This is an internal function of package FRTM which allows controlling the parameters to implement real-time registration step in the FRTM method.

Usage

par.rtr(
  seq_x = NULL,
  delta_d = 0.05,
  delta_v = 0.03,
  delta_c = 0.04,
  Delta = 0.1,
  length_grid_window = 10,
  length_grid_owindow = 20,
  eval_seq_der = seq(0.02, 0.1, length.out = 10),
  perc_basis_x_reg = 0.3
)

Arguments

Author(s)

F. Centofanti

See Also

FRTM_PhaseI

Examples

library(funcharts)
par.rtr()

Multivariate functional principal components analysis

Description

Multivariate functional principal components analysis (MFPCA) performed on an object of class mfd. It is a wrapper to fda::pca.fd, providing some additional arguments.

Usage

pca_mfd(mfdobj, scale = TRUE, nharm = 20)

Arguments

Value

Modified pca.fd object, with multivariate functional principal component scores summed over variables (fda::pca.fd returns an array of scores when providing a multivariate functional data object). Moreover, the multivariate functional principal components given in harmonics are converted to the mfd class.

See Also

scale_mfd

Examples

library(funcharts)
mfdobj <- data_sim_mfd()
pca_obj <- pca_mfd(mfdobj)
plot_pca_mfd(pca_obj)

Get a list of multivariate functional principal component analysis models estimated on functional data each evolving up to an intermediate domain point.

Description

This function produces a list of objects, each of them contains the result of applying pca_mfd to a multivariate functional data object evolved up to an intermediate domain point.

Usage

pca_mfd_real_time(mfdobj_list, scale = TRUE, nharm = 20, ncores = 1)

Arguments

Value

A list of lists each produced by pca_mfd, corresponding to a given instant.

Author(s)

C. Capezza

See Also

pca_mfd

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
mfdobj_list <- get_mfd_list_real_time(air[c("CO", "temperature")],
                                      n_basis = 15,
                                      lambda = 1e-2,
                                      k_seq = seq(0.25, 1, length = 5))
mod_list <- pca_mfd_real_time(mfdobj_list)

Plot the results of the Phase I and the Phase II of the AMFCC

Description

This function provides plots of either the monitoring statistics or the contribution plot for a given observation.

Usage

## S3 method for class 'AMFCC_PhaseI'
plot(x, ...)

## S3 method for class 'AMFCC_PhaseII'
plot(x, ...)

Arguments

Value

No return value, called for side effects.

Examples

library(funcharts)
N <- 10
l_grid <- 10
p <- 2
grid <- seq(0, 1, l = l_grid)


Xall_tra <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  correlation_type_x = c("Bessel", "Gaussian")
)
X_tra <-
  data.frame(
    x = c(Xall_tra$X_list[[1]], Xall_tra$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

Xall_II <- funcharts::simulate_mfd(
  nobs = N,
  p = p,
  ngrid = l_grid,
  shift_type_x = list("A", "B"),
  d_x = c(10, 10),
  correlation_type_x = c("Bessel", "Gaussian")
)

X_II <-
  data.frame(
    x = c(Xall_II$X_list[[1]], Xall_II$X_list[[2]]),
    timeindex = rep(rep(1:l_grid, each = (N)), p),
    curve = rep(1:(N), l_grid * p),
    var = rep(1:p, each = l_grid * N)
  )

# AMFCC -------------------------------------------------------------------
print("AMFCC")

mod_phaseI_AMFCC <- AMFCC_PhaseI(
  data_tra = X_tra,
  data_tun =
    NULL,
  grid = grid,
  ncores = 1
)

mod_phaseII_AMFCC <- AMFCC_PhaseII(data = X_II,
mod_Phase_I = mod_phaseI_AMFCC,
ncores = 1)

plot(mod_phaseII_AMFCC)
plot(mod_phaseII_AMFCC,type='cont',ind_obs=1)

Plot the results of the Phase I and the Phase II of the FRTM

Description

This function provides plots of the Hotelling's T^{2} and SPE control charts.

Usage

## S3 method for class 'FRTM_PhaseI'
plot(x, ...)

## S3 method for class 'FRTM_PhaseII'
plot(x, ...)

Arguments

Value

No return value, called for side effects.

Examples

library(funcharts)
data <- simulate_data_FRTM(n_obs = 20)

data_oc <-
  simulate_data_FRTM(
    n_obs = 2,
    scenario = "1",
    shift = "OC_h",
    severity = 0.5
  )

lambda <- 10 ^ -5
max_x <- max(unlist(data$grid_i))
seq_t_tot <- seq(0, 1, length.out = 30)[-1]
seq_x <- seq(0.1, max_x, length.out = 10)


## Not run: 
  mod_phaseI_FRTM <- FRTM_PhaseI(
    data_tra =  data,
    control.FDTW = list(
      M = 30,
      N = 30,
      lambda = lambda,
      seq_t = seq_t_tot,
      iter_tem = 1,
      iter = 1
    ),
    control.rtr = list(seq_x = seq_x)
  )
  mod_phaseII_FRTM <- FRTM_PhaseII(data_oc = data_oc , mod_phaseI = mod_phaseI_FRTM)

  plot(mod_phaseI_FRTM)
  plot(mod_phaseII_FRTM)

## End(Not run)

Plot the results of the Mixed Functional Principal Component Analysis (mFPCA)

Description

This function provides plots of the principal components of the mFPCA.

Usage

## S3 method for class 'mFPCA'
plot(x, ...)

Arguments

Value

No return value, called for side effects.

Examples

library(funcharts)

data <- simulate_data_FRTM(n_obs = 100)
X <- sapply(1:100, function(ii)
  data$x_true[[ii]])
x_fd <-
  fda::smooth.basis(y = X,
                    argvals =  data$grid,
                    fda::create.bspline.basis(c(0, 1), 30))$fd
H <- sapply(1:100, function(ii)
  data$h[[ii]])
h_fd <-
  fda::smooth.basis(y = H,
                    argvals =  data$grid,
                    fda::create.bspline.basis(c(0, 1), 30))$fd
mod_mFPCA <- mFPCA(x_fd, h_fd, ncom = "ptv", par_ncom = 0.95)
plot(mod_mFPCA)

Plot a Bivariate Functional Data Object.

Description

Plot an object of class bifd using ggplot2 and geom_tile. The object must contain only one single functional replication.

Usage

plot_bifd(bifd_obj, type_plot = "raster", phi = 40, theta = 40)

Arguments

Value

A ggplot with a geom_tile layer providing a plot of the bivariate functional data object as a heat map.

Examples

library(funcharts)
mfdobj <- data_sim_mfd(nobs = 1)
tp <- tensor_product_mfd(mfdobj)
plot_bifd(tp)

Plot bootstrapped estimates of the scalar-on-function regression coefficient

Description

Plot bootstrapped estimates of the scalar-on-function regression coefficient for empirical uncertainty quantification. For each iteration, a data set is sampled with replacement from the training data use to fit the model, and the regression coefficient is estimated.

Usage

plot_bootstrap_sof_pc(mod, nboot = 25, ncores = 1)

Arguments

Value

A ggplot showing several bootstrap replicates of the multivariate functional coefficients estimated fitting the scalar-on-function linear model. Gray lines indicate the different bootstrap estimates, the black line indicate the estimate on the entire dataset.

Author(s)

C. Capezza

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
y <- rowMeans(air$NO2)
mod <- sof_pc(y, mfdobj_x)
plot_bootstrap_sof_pc(mod, nboot = 5)

Plot control charts

Description

This function takes as input a data frame produced with functions such as control_charts_pca and 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.

Usage

plot_control_charts(cclist, nobsI = 0)

Arguments

Details

Out-of-control points are signaled by colouring them in red.

Value

A ggplot with the functional control charts.

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 real-time control charts

Description

This function produces a ggplot with the desired real-time control charts. It takes as input a list of data frames, produced with functions such as regr_cc_fof_real_time and control_charts_sof_pc_real_time, and the id of the observations for which real-time control charts are desired to be plotted. For each control chart, the solid line corresponds to the profile of the monitoring statistic and it is compared against control limits plotted as dashed lines. If a line is outside its limits it is coloured in red.

Usage

plot_control_charts_real_time(cclist, id_num)

Arguments

Details

If the line, representing the profile of the monitoring statistic over the functional domain, is out-of-control, then it is coloured in red.

Value

A ggplot with the real-time functional control charts.

See Also

regr_cc_fof_real_time, control_charts_sof_pc_real_time

Examples

library(funcharts)
data("air")
air1 <- lapply(air, function(x) x[1:8, , drop = FALSE])
air2 <- lapply(air, function(x) x[9:10, , drop = FALSE])
mfdobj_x1_list <- get_mfd_list_real_time(air1[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_x2_list <- get_mfd_list_real_time(air2[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
y1 <- rowMeans(air1$NO2)
y2 <- rowMeans(air2$NO2)
mod_list <- sof_pc_real_time(y1, mfdobj_x1_list)
cclist <- regr_cc_sof_real_time(
  mod_list = mod_list,
  y_new = y2,
  mfdobj_x_new = mfdobj_x2_list,
  include_covariates = TRUE)
plot_control_charts_real_time(cclist, 1)

Plot a Multivariate Functional Data Object.

Description

Plot an object of class mfd using ggplot2 and patchwork.

Usage

plot_mfd(
  mfdobj,
  mapping = NULL,
  data = NULL,
  stat = "identity",
  position = "identity",
  na.rm = TRUE,
  orientation = NA,
  show.legend = NA,
  inherit.aes = TRUE,
  type_mfd = "mfd",
  y_lim_equal = FALSE,
  ...
)

Arguments

Value

A plot of the multivariate functional data object.

Examples

library(funcharts)
library(ggplot2)
mfdobj <- data_sim_mfd()
ids <- mfdobj$fdnames[[2]]
df <- data.frame(id = ids, first_two_obs = ids %in% c("rep1", "rep2"))
plot_mfd(mapping = aes(colour = first_two_obs),
         data = df,
         mfdobj = mfdobj)

Plot multivariate functional object over the training data set

Description

This function plots selected functions in a phase_II monitoring data set against the corresponding training data set to be compared.

Usage

plot_mon(cclist, fd_train, fd_test, plot_title = FALSE, print_id = FALSE)

Arguments

Value

A ggplot of the multivariate functional data. In particular, the multivariate functional data given in fd_train are plotted on the background in gray, while the multivariate functional data given in fd_test are plotted on the foreground, the colour of each curve is black or red depending on if that curve was signal as anomalous by at least a contribution plot.

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]
y_tuning <- y[61:90]
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 <- regr_cc_sof(object = mod,
                      y_new = y2,
                      mfdobj_x_new = mfdobj_x2,
                      y_tuning = y_tuning,
                      mfdobj_x_tuning = mfdobj_x_tuning,
                      include_covariates = TRUE)
get_ooc(cclist)
cont_plot(cclist, 3)
plot_mon(cclist, fd_train = mfdobj_x1, fd_test = mfdobj_x2[3])

Plot the harmonics of a pca_mfd object

Description

Plot the harmonics of a pca_mfd object

Usage

plot_pca_mfd(pca, harm = 0, scaled = FALSE)

Arguments

Value

A ggplot of the harmonics/multivariate functional principal components contained in the object pca.

Examples

library(funcharts)
mfdobj <- data_sim_mfd()
pca_obj <- pca_mfd(mfdobj)
plot_pca_mfd(pca_obj)

Use a function-on-function linear regression model for prediction

Description

Predict new observations of the functional response variable and calculate the corresponding prediction error (and their standardized or studentized version) given new observations of functional covariates and a fitted function-on-function linear regression model.

Usage

predict_fof_pc(object, mfdobj_y_new, mfdobj_x_new)

Arguments

Value

A list of mfd objects. It contains:

• pred_error: the prediction error of the standardized functional response variable,

• pred_error_original_scale: the prediction error of the functional response variable on the original scale,

• y_hat_new: the prediction of the functional response observations on the original scale,

• y_z_new: the standardized version of the functional response observations provided in mfdobj_y_new,

• y_hat_z_new: the prediction of the functional response observations on the standardized/studentized scale.

Author(s)

C. Capezza, F. Centofanti

References

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3):281–294. doi:10.1080/00401706.2020.1753581

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
mfdobj_y <- get_mfd_list(air["NO2"], lambda = 1e-2)
mod <- fof_pc(mfdobj_y, mfdobj_x)
predict_fof_pc(mod,
               mfdobj_y_new = mfdobj_y,
               mfdobj_x_new = mfdobj_x)

Use a scalar-on-function linear regression model for prediction

Description

Predict new observations of the scalar response variable and calculate the corresponding prediction error, with prediction interval limits, given new observations of functional covariates and a fitted scalar-on-function linear regression model

Usage

predict_sof_pc(
  object,
  y_new = NULL,
  mfdobj_x_new = NULL,
  alpha = 0.05,
  newdata
)

Arguments

Value

A data.frame with as many rows as the number of functional replications in newdata, with the following columns:

• fit: the predictions of the response variable corresponding to new_data,

• lwr: lower limit of the 1-alpha prediction interval on the response, based on the assumption that it is normally distributed.

• upr: upper limit of the 1-alpha prediction interval on the response, based on the assumption that it is normally distributed.

• res: the residuals obtained as the values of y_new minus their fitted values. If the scalar-on-function model has been fitted with type_residual == "studentized", then the studentized residuals are calculated.

Author(s)

C. Capezza

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
y <- rowMeans(air$NO2)
mod <- sof_pc(y, mfdobj_x)
predict_sof_pc(mod)

Bind replications of two Multivariate Functional Data Objects

Description

Bind replications of two Multivariate Functional Data Objects

Usage

rbind_mfd(mfdobj1, mfdobj2)

Arguments

Value

An object of class mfd, whose variables are the same of mfdobj1 and mfdobj2 and whose replications are the union of the replications in mfdobj1 and mfdobj2.

Examples

library(funcharts)
mfdobj1 <- data_sim_mfd(nvar = 3, nobs = 4)
mfdobj2 <- data_sim_mfd(nvar = 3, nobs = 5)
dimnames(mfdobj2$coefs)[[2]] <-
  mfdobj2$fdnames[[2]] <-
  c("rep11", "rep12", "rep13", "rep14", "rep15")
mfdobj_rbind <- rbind_mfd(mfdobj1, mfdobj2)
plot_mfd(mfdobj_rbind)

Functional Regression Control Chart

Description

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.

Usage

regr_cc_fof(
  object,
  mfdobj_y_new,
  mfdobj_x_new,
  mfdobj_y_tuning = NULL,
  mfdobj_x_tuning = NULL,
  alpha = 0.05,
  include_covariates = FALSE,
  absolute_error = FALSE
)

Arguments

Value

A data.frame containing the output of the function control_charts_pca applied to the prediction errors.

Author(s)

F. Centofanti

References

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3):281–294. doi:10.1080/00401706.2020.1753581

See Also

control_charts_pca

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)

Real-time functional regression control chart

Description

This function produces a list of data frames, each of them is produced by regr_cc_fof and is needed to plot control charts for monitoring in real time a functional quality characteristic adjusted for by the effect of multivariate functional covariates.

Usage

regr_cc_fof_real_time(
  mod_list,
  mfdobj_y_new_list,
  mfdobj_x_new_list,
  mfdobj_y_tuning_list = NULL,
  mfdobj_x_tuning_list = NULL,
  alpha = 0.05,
  include_covariates = FALSE,
  absolute_error = FALSE,
  ncores = 1
)

Arguments

Value

A list of data.frames each produced by regr_cc_fof, corresponding to a given instant.

See Also

fof_pc_real_time, regr_cc_fof

Examples

library(funcharts)
data("air")
air1 <- lapply(air, function(x) x[1:8, , drop = FALSE])
air2 <- lapply(air, function(x) x[9:10, , drop = FALSE])
mfdobj_x1_list <- get_mfd_list_real_time(air1[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_x2_list <- get_mfd_list_real_time(air2[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_y1_list <- get_mfd_list_real_time(air1["NO2"],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_y2_list <- get_mfd_list_real_time(air2["NO2"],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mod_list <- fof_pc_real_time(mfdobj_y1_list, mfdobj_x1_list)
cclist <- regr_cc_fof_real_time(
  mod_list = mod_list,
  mfdobj_y_new_list = mfdobj_y2_list,
  mfdobj_x_new_list = mfdobj_x2_list)
plot_control_charts_real_time(cclist, 1)

Scalar-on-Function Regression Control Chart

Description

This function is deprecated. Use 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 include_covariates is TRUE, it also plots the Hotelling's T2 and squared prediction error control charts built on the multivariate functional covariates.

Usage

regr_cc_sof(
  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
)

Arguments

Details

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.

Value

A data.frame with as many rows as the number of functional replications in mfdobj_x_new, with the following columns:

• 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.

Author(s)

C. Capezza

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

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)

Real-time Scalar-on-Function Regression Control Chart

Description

This function builds a list of data frames, each of them is produced by regr_cc_sof and is needed to plot control charts for monitoring in real time a scalar quality characteristic adjusted for by the effect of 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.

Usage

regr_cc_sof_real_time(
  mod_list,
  y_new,
  mfdobj_x_new_list,
  y_tuning = NULL,
  mfdobj_x_tuning_list = NULL,
  alpha = 0.05,
  parametric_limits = TRUE,
  include_covariates = FALSE,
  absolute_error = FALSE,
  ncores = 1
)

Arguments

Value

A list of data.frames each produced by regr_cc_sof, corresponding to a given instant.

See Also

sof_pc_real_time, regr_cc_sof

Examples

library(funcharts)
data("air")
air1 <- lapply(air, function(x) x[1:8, , drop = FALSE])
air2 <- lapply(air, function(x) x[9:10, , drop = FALSE])
mfdobj_x1_list <- get_mfd_list_real_time(air1[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_x2_list <- get_mfd_list_real_time(air2[c("CO", "temperature")],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_y1_list <- get_mfd_list_real_time(air1["NO2"],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mfdobj_y2_list <- get_mfd_list_real_time(air2["NO2"],
                                         n_basis = 15,
                                         lambda = 1e-2,
                                         k_seq = c(0.5, 1))
mod_list <- fof_pc_real_time(mfdobj_y1_list, mfdobj_x1_list)
cclist <- regr_cc_fof_real_time(
  mod_list = mod_list,
  mfdobj_y_new_list = mfdobj_y2_list,
  mfdobj_x_new_list = mfdobj_x2_list)
plot_control_charts_real_time(cclist, 1)

Robust multivariate functional principal components analysis

Description

It performs robust MFPCA as described in Capezza et al. (2024).

Usage

rpca_mfd(
  mfdobj,
  center = "fusem",
  scale = "funmad",
  nharm = 20,
  method = "ROBPCA",
  alpha = 0.8
)

Arguments

Value

An object of pca_mfd class, as returned by the pca_mfd function when performing non robust multivariate functional principal component analysis.

Author(s)

C. Capezza, F. Centofanti

References

Capezza, C., Centofanti, F., Lepore, A., Palumbo, B. (2024) Robust Multivariate Functional Control Chart. Technometrics, 66(4):531–547, doi:10.1080/00401706.2024.2327346.

Centofanti, F., Colosimo, B.M., Grasso, M.L., Menafoglio, A., Palumbo, B., Vantini, S. (2023) Robust functional ANOVA with application to additive manufacturing. Journal of the Royal Statistical Society Series C: Applied Statistics 72(5), 1210–1234 doi:10.1093/jrsssc/qlad074

Croux, C., Ruiz-Gazen, A. (2005). High breakdown estimators for principal components: The projection-pursuit approach revisited. Journal of Multivariate Analysis, 95, 206–226, doi:10.1016/j.jmva.2004.08.002.

Hubert, M., Rousseeuw, P.J., Branden, K.V. (2005) ROBPCA: A New Approach to Robust Principal Component Analysis, Technometrics 47(1), 64–79, doi:10.1198/004017004000000563

Locantore, N., Marron, J., Simpson, D., Tripoli, N., Zhang, J., Cohen K., K. (1999), Robust principal components for functional data. Test, 8, 1-28. doi:10.1007/BF02595862

Examples

library(funcharts)
dat <- simulate_mfd(nobs = 20, p = 1, correlation_type_x = "Bessel")
mfdobj <- get_mfd_list(dat$X_list, n_basis = 5)

# contaminate first observation
mfdobj$coefs[, 1, ] <- mfdobj$coefs[, 1, ] + 0.05

# plot_mfd(mfdobj) # plot functions to see the outlier
# pca <- pca_mfd(mfdobj) # non robust MFPCA
rpca <- rpca_mfd(mfdobj) # robust MFPCA
# plot_pca_mfd(pca, harm = 1) # plot first eigenfunction, affected by outlier
# plot_pca_mfd(rpca, harm = 1) # plot first eigenfunction in robust case

Standardize Multivariate Functional Data.

Description

Scale multivariate functional data contained in an object of class mfd by subtracting the mean function and dividing by the standard deviation function.

Usage

scale_mfd(mfdobj, center = TRUE, scale = TRUE)

Arguments

Details

This function has been written to work similarly as the function scale for matrices. When calculated, attributes center and scale are of class fd and have the same structure you get when you use fda::mean.fd and fda::sd.fd.

Value

A standardized object of class mfd, with two attributes, if calculated, center and scale, storing the mean and standard deviation functions used for standardization.

Examples

library(funcharts)
mfdobj <- data_sim_mfd()
mfdobj_scaled <- scale_mfd(mfdobj)

Simulate example data for funcharts

Description

Function used to simulate three data sets to illustrate the use of funcharts. It uses the function simulate_mfd, which creates a data set with three functional covariates, a functional response generated as a function of the three functional covariates, and a scalar response generated as a function of the three functional covariates. This function generates three data sets, one for phase I, one for tuning, i.e., to estimate the control chart limits, and one for phase II monitoring. see also simulate_mfd.

Usage

sim_funcharts(nobs1 = 1000, nobs_tun = 1000, nobs2 = 60)

Arguments

Value

A list with three objects, datI contains the phase I data, datI_tun contains the tuning data, datII contains the phase II data. In the phase II data, the first group of observations are in control, the second group of observations contains a moderate mean shift, while the third group of observations contains a severe mean shift. The shift types are described in the paper from Capezza et al. (2023).

References

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3):281–294. doi:10.1080/00401706.2020.1753581

Capezza, C., Centofanti, F., Lepore, A., Menafoglio, A., Palumbo, B., & Vantini, S. (2023). funcharts: Control charts for multivariate functional data in R. Journal of Quality Technology, 55(5), 566–583. doi:10.1080/00224065.2023.2219012

Simulate data for real-time monitoring of univariate functional data

Description

Generate synthetic data as in the simulation study of Centofanti et al. (2024).

Usage

simulate_data_FRTM(
  n_obs = 100,
  scenario = "1",
  shift = "IC",
  alignemnt_level = "M1",
  t_out_type = "0.3",
  severity = 0.5,
  grid = seq(0, 1, length.out = 100)
)

Arguments

Value

A list containing the following arguments:

x_err: A list containing the discrete observations for each curve.

grid_i: A list of vector of time points where the curves are sampled.

h: A list containing the discrete observations of the warping function for each curve.

template: The discrete observations of the true template function.

grid_template: Time points where the template is sampled.

x_true: A list containing the discrete observations of the amplitude function for each curve.

grid: Grid of evaluation points.

out_control_t: Time of the change point.

References

Centofanti, F., A. Lepore, M. Kulahci, and M. P. Spooner (2024). Real-time monitoring of functional data. Accepted for publication in Journal of Quality Technology.

Examples

library(funcharts)
data<-simulate_data_FRTM(n_obs=20)

Simulate a data set for funcharts

Description

Function used to simulate a data set to illustrate the use of funcharts. By default, it creates a data set with three functional covariates, a functional response generated as a function of the three functional covariates through a function-on-function linear model, and a scalar response generated as a function of the three functional covariates through a scalar-on-function linear model. This function covers the simulation study in Centofanti et al. (2021) for the function-on-function case and also simulates data in a similar way for the scalar response case. It is possible to select the number of functional covariates, the correlation function type for each functional covariate and the functional response, moreover it is possible to provide manually the mean and variance functions for both functional covariates and the response. In the default case, the function generates in-control data. Additional arguments can be used to generate additional data that are out of control, with mean shifts according to the scenarios proposed by Centofanti et al. (2021). Each simulated observation of a functional variable consists of a vector of discrete points equally spaced between 0 and 1 (by default 150 points), generated with noise.

Usage

simulate_mfd(
  nobs = 1000,
  p = 3,
  R2 = 0.97,
  shift_type_y = "0",
  shift_type_x = c("0", "0", "0"),
  correlation_type_y = "Bessel",
  correlation_type_x = c("Bessel", "Gaussian", "Exponential"),
  d_y = 0,
  d_y_scalar = 0,
  d_x = c(0, 0, 0),
  n_comp_y = 10,
  n_comp_x = 50,
  P = 500,
  ngrid = 150,
  save_beta = FALSE,
  mean_y = NULL,
  mean_x = NULL,
  variance_y = NULL,
  variance_x = NULL,
  sd_y = 0.3,
  sd_x = c(0.3, 0.05, 0.3),
  seed
)

Arguments

Value

A list with the following elements:

• X_list is a list of p matrices, each with dimension nobsxngrid, containing the simulated observations of the multivariate functional covariate

• Y is a nobsxngrid matrix with the simulated observations of the functional response

• y_scalar is a vector of length nobs with the simulated observations of the scalar response

• beta_fof, if save_beta = TRUE, is a list of p matrices, each with dimension PxP with the discretized functional coefficients of the function-on-function regression

• beta_sof, if save_beta = TRUE, is a list of p vectors, each with length P, with the discretized functional coefficients of the scalar-on-function regression

References

Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3):281–294. doi:10.1080/00401706.2020.1753581

Scalar-on-function linear regression based on principal components

Description

Scalar-on-function linear regression based on principal components. This function performs multivariate functional principal component analysis (MFPCA) to extract multivariate functional principal components from the multivariate functional covariates, then it builds a linear regression model of a scalar response variable on the covariate scores. Functional covariates are standardized before the regression. See Capezza et al. (2020) for additional details.

Usage

sof_pc(
  y,
  mfdobj_x,
  tot_variance_explained = 0.9,
  selection = "variance",
  single_min_variance_explained = 0,
  components = NULL
)

Arguments

Value

a list containing the following arguments:

• mod: an object of class lm that is a linear regression model where the scalar response variable is y and the covariates are the MFPCA scores of the functional covariates,

• mod$coefficients contains the matrix of coefficients of the functional regression basis functions,

• pca: an object of class pca_mfd obtained by doing MFPCA on the functional covariates,

• beta_fd: an object of class mfd object containing the functional regression coefficient \beta(t) estimated with the scalar-on-function linear regression model,

• components: a vector of integers with the components selected in the pca model,

• selection: the same as the provided argument

• single_min_variance_explained: the same as the provided argument

• tot_variance_explained: the same as the provided argument

• gcv: a vector whose j-th element is the GCV score obtained when retaining the first j components in the MFPCA model.

• PRESS: a vector whose j-th element is the PRESS statistic obtained when retaining the first j components in the MFPCA model.

Author(s)

C. Capezza

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

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
fun_covariates <- c("CO", "temperature")
mfdobj_x <- get_mfd_list(air[fun_covariates], lambda = 1e-2)
y <- rowMeans(air$NO2)
mod <- sof_pc(y, mfdobj_x)

Get a list of scalar-on-function linear regression models estimated on functional data each evolving up to an intermediate domain point.

Description

This function produces a list of objects, each of them contains the result of applying sof_pc to a scalar response variable and multivariate functional covariates evolved up to an intermediate domain point. See Capezza et al. (2020) for additional details on real-time monitoring.

Usage

sof_pc_real_time(
  y,
  mfd_real_time_list,
  single_min_variance_explained = 0,
  tot_variance_explained = 0.9,
  selection = "PRESS",
  components = NULL,
  ncores = 1
)

Arguments

Value

A list of lists each produced by sof_pc, corresponding to a given instant.

Author(s)

C. Capezza

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

See Also

sof_pc, get_mfd_df_real_time, get_mfd_list_real_time

Examples

library(funcharts)
data("air")
air <- lapply(air, function(x) x[1:10, , drop = FALSE])
mfdobj_list <- get_mfd_list_real_time(air[c("CO", "temperature")],
                                      n_basis = 15,
                                      lambda = 1e-2,
                                      k_seq = c(0.5, 0.75, 1))
y <- rowMeans(air$NO2)
mod_list <- sof_pc_real_time(y, mfdobj_list)

Tensor product of two Multivariate Functional Data objects

Description

This function returns the tensor product of two Multivariate Functional Data objects. Each object must contain only one replication.

Usage

tensor_product_mfd(mfdobj1, mfdobj2 = NULL)

Arguments

Value

An object of class bifd. If we denote with x(s)=(x_1(s),...,x_p(s)) the vector of p functions represented by mfdobj1 and with y(t)=(y_1(t),...,y_q(t)) the vector of q functions represented by mfdobj2, the output is the vector of pq bivariate functions

f(s,t)=(x_1(s)y_1(t),...,x_1(s)y_q(t), ...,x_p(s)y_1(t),...,x_p(s)y_q(t)).

Examples

library(funcharts)
mfdobj1 <- data_sim_mfd(nobs = 1, nvar = 3)
mfdobj2 <- data_sim_mfd(nobs = 1, nvar = 2)
tensor_product_mfd(mfdobj1)
tensor_product_mfd(mfdobj1, mfdobj2)

Get the index of the out of control observations from control charts

Description

This function returns a list for each control chart and returns the id of all observations that are out of control in that control chart.

Usage

which_ooc(cclist)

Arguments

Value

A list of as many data.frame objects as the control charts in cclist. Each data frame has two columns, the n contains an index number giving the observation in the phase II data set, i.e. 1 for the first observation, 2 for the second, and so on, while the id column contains the id of the observation, which can be general and depends on the specific data set.

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]
y_tuning <- y[61:90]
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 <- regr_cc_sof(object = mod,
                      y_new = y2,
                      mfdobj_x_new = mfdobj_x2,
                      y_tuning = y_tuning,
                      mfdobj_x_tuning = mfdobj_x_tuning,
                      include_covariates = TRUE)
which_ooc(cclist)