| Title: | Copula-Based Estimation and Statistical Process Control for Serially Correlated Time Series |
|---|---|
| Description: | Estimation and statistical process control are performed under copula-based time-series models. Available are statistical methods in Long and Emura (2014 JCSA), Emura et al. (2017 Commun Stat-Simul) <DOI:10.1080/03610918.2015.1073303>, Huang and Emura (2021 Commun Stat-Simul) <DOI:10.1080/03610918.2019.1602647>, Lin et al. (2021 Comm Stat-Simul) <DOI:10.1080/03610918.2019.1652318>, Sun et al. (2020 JSS Series in Statistics)<DOI:10.1007/978-981-15-4998-4>, and Huang and Emura (2021, in revision). |
| Authors: | Takeshi Emura, Xinwei Huang, Ting-Hsuan Long, Li-Hsien Sun |
| Maintainer: | Takeshi Emura <[email protected]> |
| License: | GPL-2 |
| Version: | 2.9 |
| Built: | 2025-02-07 06:47:26 UTC |
| Source: | CRAN |
Copulas are applied to model a Markov dependence for serially correlated time series. The Clayton and Joe copulas are available to specify the dependence structure. The normal and binomial distributions are available for the marginal model. Maximum likelihood estimation is implmented for estimating parameters, and a Shewhart control chart is drawn for performing statistical process control.
| Package: | Copula.Markov |
| Type: | Package |
| Version: | 2.9 |
| Date: | 2021-11-29 |
| License: GPL-2 |
Emura T, Huang XW, Chen WR, Long TH, Sun LH. Maintainer: Takeshi Emura <[email protected]>
Chen W (2018) Copula-based Markov chain model with binomial data, NCU Library
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46(4):3067-87
Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association, 52(4):466-96
Lin WC, Emura T, Sun LH (2021), Estimation under copula-based Markov normal mixture models for serially correlated data, Communications in Statistics - Simulation and Computation, 50(12):4483-515
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
Time-series data are generated under a copula-based Markov chain model with the Clayton copula. See Long et al. (2014) and Emura et al. (2017) for the details.
Clayton.Markov.DATA(n, mu, sigma, alpha)Clayton.Markov.DATA(n, mu, sigma, alpha)
n |
sample size |
mu |
mean |
sigma |
standard deviation |
alpha |
association parameter |
-1<alpha<0 for negative association; alpha>0 for positive association
Time series data of size n.
Takeshi Emura
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87
Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96
set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=8) Clayton.Markov.MLE(Y,plot=TRUE)set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=8) Clayton.Markov.MLE(Y,plot=TRUE)
Time-series data are generated under a copula-based Markov chain model with the Clayton copula and binomial margin.
Clayton.Markov.DATA.binom(n, size, prob, alpha)Clayton.Markov.DATA.binom(n, size, prob, alpha)
n |
number of observations |
size |
number of binomial trials |
prob |
binomial probability; 0<p<1 |
alpha |
association parameter |
-1<alpha<0 for negative association; alpha>0 for positive association
Time series data of size n (this is different from the number of binomial trials = "size").
Huang XW, Chen W, Emura T
Chen W (2018) Copula-based Markov chain model with binomial data, NCU Library
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) ### sample mean and SD ### mean(Y) sd(Y) ### true mean and SD ### size*prob sqrt(size*prob*(1-prob))size=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) ### sample mean and SD ### mean(Y) sd(Y) ### true mean and SD ### size*prob sqrt(size*prob*(1-prob))
Perform a parametric bootstrap test based on the Cramer-von Mises and Kolmogorov-Smirnov statistics as proposed by Huang and Emura (2019).
Clayton.Markov.GOF(Y, k = 3, D = 1, B = 200,GOF.plot=FALSE, method = "Newton")Clayton.Markov.GOF(Y, k = 3, D = 1, B = 200,GOF.plot=FALSE, method = "Newton")
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
D |
diameter for U(-D, D) used in randomized Newton-Raphson |
B |
the number of Bootstrap replications |
GOF.plot |
if TRUE, show the model diagnostic plots for B bootstrap replications |
method |
Newton-Raphson method or nlm can be chosen |
CM |
The Cramer-von Mises statistic and its P-value |
KS |
The Kolmogorov-Smirnov statistic and its P-value |
CM.boot |
Bootstrap values of the Cramer-von Mises statistics |
KS.boot |
Bootstrap values of the Kolmogorov-Smirnov statistics |
Takeshi Emura
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87
Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=2) Clayton.Markov.GOF(Y,B=5,GOF.plot=TRUE)set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=2) Clayton.Markov.GOF(Y,B=5,GOF.plot=TRUE)
Perform a parametric bootstrap test based on the Cramer-von Mises and Kolmogorov-Smirnov statistics as proposed by Huang and Emura (2019) and Huang et al. (2019-).
Clayton.Markov.GOF.binom(Y, k = 3, size, B = 200,GOF.plot=FALSE, method = "Newton")Clayton.Markov.GOF.binom(Y, k = 3, size, B = 200,GOF.plot=FALSE, method = "Newton")
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
size |
number of binomial trials |
B |
the number of Bootstrap replications |
GOF.plot |
if TRUE, show the model diagnostic plots for B bootstrap replications |
method |
Newton-Raphson method or nlm can be chosen |
CM |
The Cramer-von Mises statistic and its P-value |
KS |
The Kolmogorov-Smirnov statistic and its P-value |
CM.boot |
Bootstrap values of the Cramer-von Mises statistics |
KS.boot |
Bootstrap values of the Kolmogorov-Smirnov statistics |
Huang XW, Emura T
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) Clayton.Markov.GOF.binom(Y,size=size,B=5,k=3,GOF.plot=TRUE) ## B=5 to save timesize=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) Clayton.Markov.GOF.binom(Y,size=size,B=5,k=3,GOF.plot=TRUE) ## B=5 to save time
The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Clayton copula and the marginal (stationary) distribution follows the normal distribution.
Clayton.Markov.MLE(Y, k = 3, D = 1, plot = TRUE,GOF=FALSE,method = "nlm")Clayton.Markov.MLE(Y, k = 3, D = 1, plot = TRUE,GOF=FALSE,method = "nlm")
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
D |
diameter for U(-D, D) used in randomized Newton-Raphson |
plot |
show the control chart if TRUE |
GOF |
show the model diagnostic plot if TRUE |
method |
apply "nlm" or "Newton" method |
mu |
estimate, SE, and 95 percent CI |
sigma |
estimate, SE, and 95 percent CI |
alpha |
estimate, SE, and 95 percent CI |
Control_Limit |
Center = mu, LCL = mu - k*sigma, UCL = mu + k*sigma |
out_of_control |
IDs for out-of-control points |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
KS.test |
KS statistics |
CM.test |
CM statistics |
log.likelihood |
Log-likelihood value for the estimation |
Long TH, Huang XW and Emura T
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87
Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96
set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=2) Clayton.Markov.MLE(Y,plot=TRUE)set.seed(1) Y=Clayton.Markov.DATA(n=1000,mu=0,sigma=1,alpha=2) Clayton.Markov.MLE(Y,plot=TRUE)
The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Clayton copula and the marginal (stationary) distribution follows the normal distribution.
Clayton.Markov.MLE.binom(Y, size, k = 3, method="nlm", plot = TRUE, GOF=FALSE)Clayton.Markov.MLE.binom(Y, size, k = 3, method="nlm", plot = TRUE, GOF=FALSE)
Y |
vector of observations |
size |
numbe of binomial trials |
method |
nlm or Newton |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
plot |
show the control chart if TRUE |
GOF |
show the model diagnostic plot if TRUE |
p |
estimate, SE, and 95 percent CI |
alpha |
estimate, SE, and 95 percent CI |
Control_Limit |
Center = n*p, LCL = mu - k*sigma, UCL = mu + k*sigma |
out_of_control |
IDs for out-of-control points |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
KS.test |
KS statistics |
CM.test |
CM statistics |
log_likelihood |
Log-likelihood value for the estimation |
Huang XW, Emura T
Chen W (2018) Copula-based Markov chain model with binomial data, NCU Library
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) Clayton.Markov.MLE.binom(Y,size=size,k=3,plot=TRUE)size=50 prob=0.5 alpha=2 set.seed(1) Y=Clayton.Markov.DATA.binom(n=500,size,prob,alpha) Clayton.Markov.MLE.binom(Y,size=size,k=3,plot=TRUE)
Time-series data are generated under a copula-based 2nd order Markov chain model with the Clayton copula.
Clayton.Markov2.DATA(n, mu, sigma, alpha)Clayton.Markov2.DATA(n, mu, sigma, alpha)
n |
sample size |
mu |
mean |
sigma |
standard deviation |
alpha |
association parameter |
-1<alpha<0 for negative association; alpha>0 for positive association
Time series data of size n
Xinwei Huang and Takeshi Emura
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
Clayton.Markov2.DATA(n = 100, mu = 0, sigma = 1, alpha = 2)Clayton.Markov2.DATA(n = 100, mu = 0, sigma = 1, alpha = 2)
The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Clayton copula and the marginal (stationary) distribution follows the normal distribution. The model diagnostic plot is also given (by the option "GOF=TRUE"). See Huang and Emura (2019) for the methodological details.
Clayton.Markov2.MLE(Y, k = 3, D = 1, plot = TRUE, GOF=FALSE)Clayton.Markov2.MLE(Y, k = 3, D = 1, plot = TRUE, GOF=FALSE)
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
D |
diameter for U(-D, D) used in randomized Newton-Raphson |
plot |
show the control chart if TRUE |
GOF |
show the model diagnostic plot if TRUE |
mu |
estimate, SE, and 95 percent CI |
sigma |
estimate, SE, and 95 percent CI |
alpha |
estimate, SE, and 95 percent CI |
Control_Limit |
Center = mu, LCL = mu - k*sigma, UCL = mu + k*sigma |
out_of_control |
IDs for out-of-control points |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
KS.test |
KS statistics |
CM.test |
CM statistics |
log.likelihood |
Log-likelihood value for the estimation |
Xinwei Huang and Takeshi Emura
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
Y = c(0.265, 0.256, 0.261, 0.261, 0.260, 0.257, 0.258, 0.263, 0.254, 0.254, 0.258, 0.256, 0.256, 0.265, 0.270, 0.267, 0.270, 0.267, 0.266, 0.271, 0.270, 0.264, 0.261, 0.264, 0.266, 0.264, 0.269, 0.268, 0.264, 0.262, 0.257, 0.255, 0.255, 0.253, 0.251, 0.254, 0.255) Clayton.Markov2.MLE(Y, k = 1, D = 1, plot = TRUE) Y=Clayton.Markov2.DATA(n=1000,mu=0,sigma=1,alpha=8) Clayton.Markov2.MLE(Y, plot=TRUE)Y = c(0.265, 0.256, 0.261, 0.261, 0.260, 0.257, 0.258, 0.263, 0.254, 0.254, 0.258, 0.256, 0.256, 0.265, 0.270, 0.267, 0.270, 0.267, 0.266, 0.271, 0.270, 0.264, 0.261, 0.264, 0.266, 0.264, 0.269, 0.268, 0.264, 0.262, 0.257, 0.255, 0.255, 0.253, 0.251, 0.254, 0.255) Clayton.Markov2.MLE(Y, k = 1, D = 1, plot = TRUE) Y=Clayton.Markov2.DATA(n=1000,mu=0,sigma=1,alpha=8) Clayton.Markov2.MLE(Y, plot=TRUE)
The maximum likelihood estimates are produced. The dependence model follows the Clayton copula and the marginal distribution follows the Mix-Normal distribution.
Clayton.MixNormal.Markov.MLE(y)Clayton.MixNormal.Markov.MLE(y)
y |
vector of datasets |
alpha |
estimate, SE, and 95 percent CI |
mu1 |
estimate, SE, and 95 percent CI |
mu2 |
estimate, SE, and 95 percent CI |
sigma1 |
estimate, SE, and 95 percent CI |
sigma2 |
estimate, SE, and 95 percent CI |
p |
estimate, SE, and 95 percent CI |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
log.likelihood |
Log-likelihood value for the estimation |
Sun LH, Huang XW
Lin WC, Emura T, Sun LH (2021), Estimation under copula-based Markov normal mixture models for serially correlated data, Communications in Statistics - Simulation and Computation, 50(12):4483-515
data(DowJones) Y=as.vector(DowJones$log_return) Clayton.MixNormal.Markov.MLE(y=Y)data(DowJones) Y=as.vector(DowJones$log_return) Clayton.MixNormal.Markov.MLE(y=Y)
The log return of weekly stock price of Dow Jones Industrial Average from 2008/1/1 to 2012/1/1.
data("DowJones")data("DowJones")
A data frame with 754 observations on the following 1 variables.
log_returna numeric vector
Lin WC, Emura T, Sun LH (2021), Estimation under copula-based Markov normal mixture models for serially correlated data, Communications in Statistics - Simulation and Computation, 50(12):4483-515
data(DowJones) DowJonesdata(DowJones) DowJones
Time-series data are generated under a copula-based Markov chain model with the Joe copula.
Joe.Markov.DATA(n, mu, sigma, alpha)Joe.Markov.DATA(n, mu, sigma, alpha)
n |
sample size |
mu |
mean |
sigma |
standard deviation |
alpha |
association parameter |
alpha>=1 for positive association
Time series data of size n
Takeshi Emura
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87
Long TS and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96
n=1000 alpha=2.856 ### Kendall's tau =0.5 ### mu=2 sigma=1 Y=Joe.Markov.DATA(n,mu,sigma,alpha) mean(Y) sd(Y) cor(Y[-1],Y[-n],method="kendall") Joe.Markov.MLE(Y,k=2)n=1000 alpha=2.856 ### Kendall's tau =0.5 ### mu=2 sigma=1 Y=Joe.Markov.DATA(n,mu,sigma,alpha) mean(Y) sd(Y) cor(Y[-1],Y[-n],method="kendall") Joe.Markov.MLE(Y,k=2)
Time-series data are generated under a copula-based Markov chain model with the Joe copula and binomial margin.
Joe.Markov.DATA.binom(n, size, prob, alpha)Joe.Markov.DATA.binom(n, size, prob, alpha)
n |
number of observations |
size |
number of binomial trials |
prob |
binomial probability; 0<p<1 |
alpha |
association parameter |
alpha>=1 for positive association
time series data
Huang X, Emura T
Chen W (2018) Copula-based Markov chain model with binomial data, NCU Library
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) ### sample mean and SD ### mean(Y) sd(Y) ### true mean and SD ### size*prob sqrt(size*prob*(1-prob))size=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) ### sample mean and SD ### mean(Y) sd(Y) ### true mean and SD ### size*prob sqrt(size*prob*(1-prob))
Perform a parametric bootstrap test based on the Cramer-von Mises and Kolmogorov-Smirnov statistics as proposed by Huang and Emura (2019) and Huang et al. (2019-).
Joe.Markov.GOF.binom(Y, k = 3, size, B = 200,GOF.plot=FALSE)Joe.Markov.GOF.binom(Y, k = 3, size, B = 200,GOF.plot=FALSE)
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
size |
number of binomial trials |
B |
the number of Bootstrap replications |
GOF.plot |
if TRUE, show the model diagnostic plots for B bootstrap replications |
CM |
The Cramer-von Mises statistic and its P-value |
KS |
The Kolmogorov-Smirnov statistic and its P-value |
CM.boot |
Bootstrap values of the Cramer-von Mises statistics |
KS.boot |
Bootstrap values of the Kolmogorov-Smirnov statistics |
Huang XW, Emura T
Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67
Huang XW, Emura T (2021-), Computational methods for a copula-based Markov chain model with a binomial time series, in review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) Joe.Markov.GOF.binom(Y,size=size,B=5,k=3,GOF.plot=TRUE) ## B=5 to save timesize=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) Joe.Markov.GOF.binom(Y,size=size,B=5,k=3,GOF.plot=TRUE) ## B=5 to save time
The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Joe copula and the marginal (stationary) distribution follows the normal distribution.
Joe.Markov.MLE(Y, k = 3, D = 1, plot = TRUE,GOF=FALSE,method = "nlm")Joe.Markov.MLE(Y, k = 3, D = 1, plot = TRUE,GOF=FALSE,method = "nlm")
Y |
vector of datasets |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
D |
diameter for U(-D, D) used in randomized Newton-Raphson |
plot |
show the control chart if TRUE |
GOF |
show the model diagnostic plot if TRUE |
method |
apply "nlm" or "Newton" method |
mu |
estimate, SE, and 95 percent CI |
sigma |
estimate, SE, and 95 percent CI |
alpha |
estimate, SE, and 95 percent CI |
Control_Limit |
Center = mu, LCL = mu - k*sigma, UCL = mu + k*sigma |
out_of_control |
IDs for out-of-control points |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
KS.test |
KS statistics |
CM.test |
CM statistics |
log.likelihood |
Log-likelihood value for the estimation |
Long TH, Huang XW and Takeshi Emura
Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87
Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96
n=1000 alpha=2.856 ### Kendall's tau =0.5 ### mu=2 sigma=1 Y=Joe.Markov.DATA(n,mu,sigma,alpha) mean(Y) sd(Y) cor(Y[-1],Y[-n],method="kendall") Joe.Markov.MLE(Y,k=2)n=1000 alpha=2.856 ### Kendall's tau =0.5 ### mu=2 sigma=1 Y=Joe.Markov.DATA(n,mu,sigma,alpha) mean(Y) sd(Y) cor(Y[-1],Y[-n],method="kendall") Joe.Markov.MLE(Y,k=2)
The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Joe copula and the marginal (stationary) distribution follows the binomial distribution.
Joe.Markov.MLE.binom(Y, size, k = 3, plot = TRUE, GOF=FALSE)Joe.Markov.MLE.binom(Y, size, k = 3, plot = TRUE, GOF=FALSE)
Y |
vector of observations |
size |
number of binomial trials |
k |
constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit) |
plot |
show the control chart if TRUE |
GOF |
show the model diagnostic plot if TRUE |
p |
estimate, SE, and 95 percent CI |
alpha |
estimate, SE, and 95 percent CI |
Control_Limit |
Center = n*p, LCL = mu - k*sigma, UCL = mu + k*sigma |
out_of_control |
IDs for out-of-control points |
Gradient |
gradients (must be zero) |
Hessian |
Hessian matrix |
Eigenvalue_Hessian |
Eigenvalues for the Hessian matrix |
KS.test |
KS statistics |
CM.test |
CM statistics |
log_likelihood |
Log-likelihood value for the estimation |
Huang XW, Emura T
Chen W (2018) Copula-based Markov chain model with binomial data, NCU Library
Huang XW, Emura T (2021+), Computational methods for a copula-based Markov chain model with a binomial time series, under review
size=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) Joe.Markov.MLE.binom(Y,size=size,k=3,plot=TRUE)size=50 prob=0.5 alpha=2 set.seed(1) Y=Joe.Markov.DATA.binom(n=500,size,prob,alpha) Joe.Markov.MLE.binom(Y,size=size,k=3,plot=TRUE)