Package 'CopCTS'

Conditional Quantile Estimation

Description

Given estiamted copula with copula parameter and specified marginal distribution, obtain the conditional qth quantile of Y_n+1 given Y1,...,Yn.

Usage

condQestCopC(tao,Yc,d,delta,copula,cop=NULL,theta=NULL,nIS=10000,
MARGIN=NULL,MARGIN.inv=NULL,...)

Arguments

tao	the desired quantile level, a numeric value between 0 and 1.
Yc	the Nx1 vector of observed responses that are subject to lower detection limit.
d	the lower detection limit.
delta	the Nx1 vector of censoring indicator with 1 indicating uncensored and 0 indicating left censored.
copula	the input copula object with copula parameter plugged in. If specified, cop and theta can be omitted.
cop	the choice of copula function. There are currently five available copula funcitons, including Clayton copula, Gaussian copula, Gumbel copula, Joe copula and Frank copula. Specify one from "Clayton","Gaussian","Gumbel","Joe" and "Frank".
theta	the copula parameter.
nIS	the size for sequential importance sampling. The default is 10000.
MARGIN	the marginal distribution of the latent time series.
MARGIN.inv	the inverse marginal distribution of the latent time series.
...	additional parameters for the marginal distribution of the latent time series.

Value

condQestCopC returns the conditional tao-th quantile of Y_n+1 given Y1,...,Yn based on the specified copula function and marginal distribution.

References

Li, F., Tang, Y. and Wang, H. (2018). Copula-Based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

Examples

set.seed(20)
Y = genLatentY(cop = "Clayton", theta = 1, N = 30)
d = -0.5
delta = (Y>d)
Yc = pmax(d,Y)
cq60.real = condQestCopC(0.6,Yc,d,delta,copula=claytonCopula(1),nIS = 50,
                        MARGIN=pnorm,MARGIN.inv=qnorm)
### Use selected copula
selCopC = selectCopC(cop.type = c("Clayton","Frank"),Yc,d,delta,nIS=50)
cq60.est = condQestCopC(0.6,Yc,d,delta,selCopC$Selected,nIS=50)

---

Pseudo maximum likelihood estimator of the copula parameter

Description

Obtains the pseudo maximum likelihood estimator of the copula parameter based on censored time series.

Usage

estCopC(cop="Gaussian",Yc,d,delta,nIS=500,jumps=NULL,MARGIN=NULL,...,interval=NULL)

Arguments

cop	the choice of copula function. There are currently five available copula funcitons, including Clayton copula, Gaussian copula, Gumbel copula, Joe copula and Frank copula. Specify one from "Clayton","Gaussian","Gumbel","Joe" and "Frank". The default is "Gaussian".
Yc	the Nx1 vector of observed response variable that is subject to lower detection limit.
d	the lower detection limit.
delta	the Nx1 vector of censoring indicator with 1 indicating uncensored and 0 indicating left censored.
nIS	the size for sequential importance sampling. The default is 500.
jumps	the Nx1 vector indicating whether each time t is a start of a new time series, which is deemed to be independent from the previous series. By default, jumps = c(1,rep(0,n-1)) indicating the data is one Markov sequence.
MARGIN	the marginal distribution function of the latent time series. The default is the empirical cdf: $\frac{1}{n+1}\sum_{t=1}^n I_{Y_t<=y}$ . MARGIN can also be specified as other existing distribution functions such as pnorm.
...	additional parameters for the marginal distribution of the latent time series.
interval	the lower and upper bound for the copula paraameter. By default, interval= c(-1,1) for Gaussian copula, c(-1,Inf) for Clayton copula, c(1,Inf) for Gumbel and Joe copula and c(-Inf,Inf) for Frank copula.

Value

estCopC returns a list of components including.

para	the pseudo maximum likelihood estimator of the copula parameter.
likelihood	the negative log-likelihood value corresponding to the estimated copula parameter.
copula	the estimated copula object, with estimated copula parameter plugged in.

References

Li, F., Tang, Y. and Wang, H. (2018). Copula-based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

Examples

### Using a simulated data for demonstration:
set.seed(20)
Y = genLatentY(cop="Clayton",1,30,MARGIN.inv = qt,df=3)
d = -1
Yc = pmax(d,Y)
delta = (Y>d)
## CopC estimator
estCopC(cop = "Clayton",Yc,d,delta,nIS = 50,interval = c(1,10))
## Omniscient estimator
estCopC(cop = "Clayton",Y,d,delta=rep(TRUE,length(Y)),interval = c(1,10))
## CopC estimator under true marginal
estCopC(cop = "Clayton",Yc,d,delta,nIS = 50,MARGIN=pt,df=3,interval = c(1,10))
### Analyze the water quality data:
attach(water)
Yc = TNH3[1:30]
delta = Delta[1:30]
jumps = Indep[1:30]
set.seed(1)
estCopC(cop="Clayton",Yc=Yc,d=0.02,delta=delta,jumps=jumps,interval = c(1,10),nIS=50)

---

Generation of data from the copula-based Markov model of order one

Description

Generate the latent response variable from the assumed copula-based Markov model in Li, Tang and Wang (2018).

Usage

genLatentY(cop="Gaussian",theta,N,MARGIN.inv=qnorm,...)

Arguments

cop	the choice of copula function. There are currently five available copula funcitons, including Clayton copula, Gaussian copula, Gumbel copula, Joe copula and Frank copula. Specify one from "Clayton","Gaussian","Gumbel","Joe" and "Frank". The default is "Gaussian".
theta	the copula parameter.
N	the length of the latent response.
MARGIN.inv	the inverse marginal distribution function of the latent time series. The default is qnorm(p,mean=0,sd=1), i.e., the standard normal marginal.
...	additional parameters for the inverse marginal distribution funcion of the latent time series.

Value

genLatentY returns a Nx1 vector of the latent response variable Y*

References

Li, F., Tang, Y. and Wang, H. (2018) Copula-based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

---

The selection of copula function

Description

Among a list of copulas, select the one that gives the estimates closest to the empirical copula function.

Usage

selectCopC(cop.type=c("Clayton","Gaussian","Gumbel","Joe","Frank"),
Yc,d,delta,nIS=500,jumps=NULL,MARGIN=NULL,...,intervals=NULL)

Arguments

cop.type	a Kx1 vector containing the candidate copulas, where K = length(cop.type) is the number of candidate copulas. There are currently five available copula funcitons, including Clayton copula, Gaussian copula, Gumbel copula, Joe copula and Frank copula. Select each by specifying a vector consisting of at least one element from c("Clayton","Gaussian","Gumbel","Joe","Frank").
Yc	the Nx1 vector of observed responses that are subject to lower detection limit.
d	the lower detection limit.
delta	the Nx1 vector of censoring indicator with 1 indicating uncensored and 0 indicating left censored.
nIS	the size for sequential importance sampling. The default is 500.
jumps	the Nx1 vector indicating whether each time t is a start of a new time series, which is deemed to be independent from the previous series.
MARGIN	the marginal distribution of the latent time series.
...	additional parameters for the marginal distribution of the latent time series.
intervals	a 2xK matrix specifying the lower and upper bound for the copula parameter of each candidate copula, where K is the number of candidate copulas.

Value

selectCopC returns a list of components including

paras	a Kx1 vector containing the estimated copula parameters for each candidate copula.
likelihoods	a Kx1 vector containing the negative log-likelihood value corresponding to the estimated copula parameter for each candidate copula.
estCop	a list containing the estimated copula object for each candidate.
L2distance	a Kx1 vector containing the L2 distance between each copula with estimated copula parameter and the empirical copula function.
Selected	The selected copula object.

References

Li, F., Tang, Y. and Wang, H. (2018) Copula-based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

See Also

estCopC.

Examples

### Example with simulated data
set.seed(20)
Y = genLatentY("Clayton",1,30,MARGIN.inv = qt,df=3)
d = -1
Yc = pmax(d,Y)
delta = (Y>d)
selectCopC(cop.type=c("Clayton","Frank"),Yc = Yc,d = d,delta = delta,nIS=50)
### Example with water data
attach(water)
Yc = TNH3[1:30]
delta = Delta[1:30]
jumps = Indep[1:30]
set.seed(1)
intv.Gaussian = c(-1,1)
intv.Clayton = c(0,20)
intv.Frank = c(0,15)
intervals = cbind(intv.Gaussian,intv.Clayton,intv.Frank)
cop.type = c("Gaussian","Clayton","Frank")
selCopC <- selectCopC(cop.type=cop.type,Yc=Yc,d=0.02,
            delta=delta,nIS = 50,jumps=jumps,intervals=intervals)
selCopC$Selected

---

Water quality (Ammonia) data

Description

This water dataset records the amount of dissolved ammonia at Susquehanna River Basin in the United States. The dissolved ammonia data were observed biweekly in Susquehanna River at Towanda, PA, from 1988 to 2014, consisting of 524 data points, with detection limit at 0.02 (mg/l).

Usage

data(water)

Format

A data frame with 524 observations on the following 4 variables.

SDate

date of measuring

TNH3

response variable, the amount of dissolved ammonia

Delta

a logical vector indicating censored as 0 and uncensored as 1

Indep

a logical vector indicating the start of a new time series that is deemed to be independent from the previous one. For the water quality data, most measurements were taken biweekly but a few have longer time gaps from the previous measurements. In our analysis of the water quality data, we treat the date that is apart from the previous measurement date more than 14 days as the start of a new independent time series.

Source

https://www.srbc.net/portals/water-quality-projects/sediment-nutrient-assessment/

References

Li, F., Tang, Y. and Wang, H. (2018) Copula-based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

Examples

data(water)
str(water)
head(water)

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