| Title: | Control Charts with Guaranteed In-Control Performance and Cautious Parameters Learning |
|---|---|
| Description: | Design and use of control charts for detecting mean changes based on a delayed updating of the in-control parameter estimates. See Capizzi and Masarotto (2019) <doi:10.1080/00224065.2019.1640096> for the description of the method. |
| Authors: | Giovanna Capizzi and Guido Masarotto |
| Maintainer: | Giovanna Capizzi <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.1 |
| Built: | 2024-10-26 06:24:39 UTC |
| Source: | CRAN |
Functions in this package allow to design, study and apply control charts based on the cautious parameter learning approach described in Capizzi and Masarotto (2019).
On system where the openMP standard is
supported, these functions can take advantage of the
computing power offered by a multicore workstation.
See omp for the default setting.
The package includes the following functions:
Computation of the control limits via stochastic
approximation: x.cl, ewma.cl,
cusum.cl;
Estimation errors and conditional run-length simulation:
ruv and rcrl;
Application to real data: cautiousLearning;
Controlling the number of used openMP cores and the random number
generator seeds: hasOMP, setOMPThreads and
setSITMOSeeds.
Giovanna Capizzi <[email protected]> and Guido Masarotto <[email protected]>
Maintainer: Giovanna Capizzi
Capizzi, G. and Masarotto, G. (2019) "Guaranteed In-Control Control Chart Performance with Cautious Parameter Learning", accepted for publication in Journal of Quality Technology, a copy of the paper can be obtained from the authors.
This function applies and, optionally, plots a control chart based on the cautious learning approach described in Capizzi and Masarotto (2019).
cautiousLearning(chart, x, mu0, s0, plot = TRUE)cautiousLearning(chart, x, mu0, s0, plot = TRUE)
chart |
list with the same elements as those returned by
|
x |
numeric vector containing the Phase II data. |
mu0, s0
|
estimates of the in-control mean and standard deviation obtained by the Phase I reference sample. |
plot |
if |
The function returns (invisibly when plot==TRUE) a numeric matrix
containing
column 1 for X and EWMA, columns 1-2 for CUSUM
|
control statistic[s] |
columns 2-4 for X and EWMA, columns 3-5 for CUSUM
|
central line, lower and upper control limits |
columns 5-7 for X and EWMA, columns 6-8 for CUSUM
|
"cautious" estimates of the mean, standard deviation and critical
value, i.e., using the notation in Capizzi and Masarotto (2019),
|
Giovanna Capizzi and Guido Masarotto
Capizzi, G. and Masarotto, G. (2019) "Guaranteed In-Control Control Chart Performance with Cautious Parameter Learning", accepted for publication in Journal of Quality Technology, a copy of the paper can be obtained from the authors.
## EWMA control chart (nominal ARL=500, ## initial estimates based on 100 in-control observations) chart <- list(chart = "EWMA", lambda = 0.2, limit = c(Linf=3.187, Delta=0.427, A=1.5, B=50, m=100)) ## Phase I estimates set.seed(12345) xic <- rnorm(100, 12 , 3) m0 <- mean(xic) s0 <- sd(xic) ## Phase II observations (one sigma mean shift starting at i=501) x <- c(rnorm(500, 12, 3), rnorm(50, 15, 3)) ## Monitoring y <- cautiousLearning(chart, x, m0, s0) head(y) tail(y)## EWMA control chart (nominal ARL=500, ## initial estimates based on 100 in-control observations) chart <- list(chart = "EWMA", lambda = 0.2, limit = c(Linf=3.187, Delta=0.427, A=1.5, B=50, m=100)) ## Phase I estimates set.seed(12345) xic <- rnorm(100, 12 , 3) m0 <- mean(xic) s0 <- sd(xic) ## Phase II observations (one sigma mean shift starting at i=501) x <- c(rnorm(500, 12, 3), rnorm(50, 15, 3)) ## Monitoring y <- cautiousLearning(chart, x, m0, s0) head(y) tail(y)
These functions compute the control limits
of X (x.cl), EWMA (ewma.cl) and CUSUM (cusum.cl)
control charts based on the cautious learning approach.
The stochastic approximation algorithm, described in
the Appendix A of Capizzi and Masarotto (2019), is used.
When openMP is supported, computation can be distribuited on multiple cores.
See omp.
x.cl(m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000) ewma.cl(lambda, m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000) cusum.cl(k, m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000)x.cl(m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000) ewma.cl(lambda, m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000) cusum.cl(k, m, arl0, alpha = 0.1, beta = 0.05, H = 200, A = 1.5, B = 50, Ninit = 1000, Nfinal = 30000)
lambda |
EWMA smoothing constant. |
k |
CUSUM reference value. |
m |
number of in-control observations used to estimate the process mean and standard deviation at the beginning of the monitoring phase. |
arl0, alpha, beta, H
|
desired in-control average run-length and constants defining the empirical guaranteed in-control performance condition. See equations (2) and (6) in Capizzi and Masarotto (2019). |
A, B
|
constants controlling when the parameters estimate are updated.
See equation (3) in Capizzi and Masarotto (2019).
If |
Ninit, Nfinal
|
number of iterations used in the stochastic approximation algorithm. See Capizzi and Masarotto (2019), Appendix A. |
A list with the following elements:
chart |
string describing the control chart ("X", "EWMA" or "CUSUM"). |
lambda |
EWMA smoothing constant (only when
|
k |
CUSUM reference value (only when
|
limit |
numeric vector of length equal to five containing the
constants defining the cautiuos learning control limits, i.e,
|
Giovanna Capizzi and Guido Masarotto
Capizzi, G. and Masarotto, G. (2019) "Guaranteed In-Control Control Chart Performance with Cautious Parameter Learning", accepted for publication in Journal of Quality Technology, a copy of the paper can be obtained from the authors.
## Only for testing: the number of iterations is reduced ## to reduce the computing time Ninit <- 50 Nfinal <- 100 H <- 50 x.cl(100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) x.cl(100, 500, A=NA, B=NA, Ninit=Ninit, Nfinal=Nfinal, H=H) ewma.cl(0.2, 100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) cusum.cl(1, 100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) ## Using the default number of iterations x.cl(100, 500) x.cl(100, 500, A=NA, B=NA) ewma.cl(0.2,100, 500) cusum.cl(1, 100, 500)## Only for testing: the number of iterations is reduced ## to reduce the computing time Ninit <- 50 Nfinal <- 100 H <- 50 x.cl(100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) x.cl(100, 500, A=NA, B=NA, Ninit=Ninit, Nfinal=Nfinal, H=H) ewma.cl(0.2, 100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) cusum.cl(1, 100, 500, Ninit=Ninit, Nfinal=Nfinal, H=H) ## Using the default number of iterations x.cl(100, 500) x.cl(100, 500, A=NA, B=NA) ewma.cl(0.2,100, 500) cusum.cl(1, 100, 500)
The functions can be used
to check if the current system supports the openMP standard;
to control the number of used cores;
to set the seeds of the random number generators.
hasOMP() setOMPThreads(nthreads) setSITMOSeeds(seed)hasOMP() setOMPThreads(nthreads) setSITMOSeeds(seed)
nthreads |
number of OpenMP threads to be used. |
seed |
number between 0 and 1 used to set the seeds of the random number generators in each threads. |
Each openMP thread (or the single thread used on systems where openMP is not
supported) uses a separate sitmo random number generator.
See sitmo-package.
Function hasOMP returns TRUE/FALSE if the system
supports/does not support openMP.
Functions setOMPThreads and setSITMOSeeds
do not return any value.
When the package is loaded, the following code is automatically executed
if (hasOMP()) setOMPThreads(parallel::detectCores())
setSITMOSeeds(runif(1))
Giovanna Capizzi and Guido Masarotto
Function ruv simulates the standardized
estimation errors at the starting of the monitoring phase
(see Section 2.3 of Capizzi and Masarotto (2019)).
Function rcrl simulates, under different in-control or
out-control scenarios, the conditional run-length given
the standardized estimation errors. When openMP is supported,
computation can be distribuited on multiple cores.
See omp.
ruv(n, m) rcrl(n, chart, u, v, tau, delta, omega, maxrl = 1000000L)ruv(n, m) rcrl(n, chart, u, v, tau, delta, omega, maxrl = 1000000L)
n |
number of simulated values. |
m |
number of in-control observations available at the starting of the monitoring phase. |
chart |
list with the same elements as those returned by
|
u, v
|
values of the estimation errors (scalars). |
tau, delta, omega
|
when i<tau, observations are distributed as N(mu,sigma^2) random variables; when i>=tau, observations are distributed as N(mu+delta*sigma, (omega*sigma)^2) random variables. |
maxrl |
run-length are truncated at i=maxrl. |
ruv |
numeric matrix of dimension nx2. |
rcrl |
integer vector of length n. |
Giovanna Capizzi and Guido Masarotto
Capizzi, G. and Masarotto, G. (2019) "Guaranteed In-Control Control Chart Performance with Cautious Parameter Learning", accepted for publication in Journal of Quality Technology, a copy of the paper can be obtained from the authors.
ruv(5, 100) ## EWMA control chart (nominal ARL=500, ## initial estimates based on 100 in-control observations) chart <- list(chart = "EWMA", lambda = 0.2, limit = c(Linf=3.187, Delta=0.427, A=1.5, B=50, m=100)) rcrl(10, chart, 2, 1, 50, 2, 1)ruv(5, 100) ## EWMA control chart (nominal ARL=500, ## initial estimates based on 100 in-control observations) chart <- list(chart = "EWMA", lambda = 0.2, limit = c(Linf=3.187, Delta=0.427, A=1.5, B=50, m=100)) rcrl(10, chart, 2, 1, 50, 2, 1)