| Title: | Copula-Based Mixed Regression Models |
|---|---|
| Description: | Estimation of 2-level factor copula-based regression models for clustered data where the response variable can be either discrete or continuous. |
| Authors: | Pavel Krupskii [aut, ctb, cph], Bouchra R. Nasri [aut, ctb, cph], Bruno N Remillard [aut, cre, cph] |
| Maintainer: | Bruno N Remillard <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.4.1 |
| Built: | 2024-10-26 06:13:31 UTC |
| Source: | CRAN |
This function computes the cdf, pdf, and associated derivatives
berncpdf(z, th)berncpdf(z, th)
z |
vector of responses |
th |
linear combination of covariates (can be negative) |
out |
Matrix of conditional cdf and pdf with derivative with respect to parameters |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = berncpdf(0,2.5)out = berncpdf(0,2.5)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
coplik(u, v, family, rot = 0, cpar, dfC = NULL, du = FALSE)coplik(u, v, family, rot = 0, cpar, dfC = NULL, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
family |
copula family: "gaussian", "t", "clayton", "frank", "fgm", "gumbel", "joe", "plackett". |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
cpar |
copula parameter |
dfC |
degrees of freedom for the Student copula (default is NULL) |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameters |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = coplik(0.3,0.5,"clayton",cpar=2,du=TRUE)out = coplik(0.3,0.5,"clayton",cpar=2,du=TRUE)
Density at (x1,x2)
dbvn(x1, x2, rh)dbvn(x1, x2, rh)
x1 |
vector of values |
x2 |
vector of values |
rh |
correlation parameter, -1< rh <1 |
out |
Vector of densities |
Pavel Krupskii
out = dbvn(0.3,0.5,-0.6)out = dbvn(0.3,0.5,-0.6)
Density at (x1,x2)
dbvn2(x1, x2, rh)dbvn2(x1, x2, rh)
x1 |
vector of values |
x2 |
vector of values |
rh |
correlation parameter, -1< rh <1 |
out |
Vector of densities |
Pavel Krupskii
out = dbvn2(0.3,0.5,-0.4)out = dbvn2(0.3,0.5,-0.4)
Density at (u,v)
dbvncop(u, v, cpar)dbvncop(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter, -1< cpar<1 |
out |
Vector of densities |
Pavel Krupskii
out = dbvncop(0.3,0.5,-0.5)out = dbvncop(0.3,0.5,-0.5)
Density at (u,v)
dbvtcop(u, v, cpar, dfC)dbvtcop(u, v, cpar, dfC)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter, -1< cpar<1 |
dfC |
degrees of freedom |
out |
Vector of densities |
Pavel Krupskii
out = dbvtcop(0.3,0.5,-0.7,25)out = dbvtcop(0.3,0.5,-0.7,25)
Evaluates the copula density at given points (u,v)#'
dcop(u, v, family, rot = 0, cpar, dfC = NULL)dcop(u, v, family, rot = 0, cpar, dfC = NULL)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
family |
copula family: "gaussian" ("normal), "t", "clayton", "frank", "fgm", "galambos", "gumbel", "joe", "huesler-reiss", "plackett". |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
cpar |
copula parameter |
dfC |
degrees of freedom for the Student copula (default is NULL) |
out |
Copula density |
out |
Vector of pdf values |
Pavel Krupskii and Bruno Remillard Mai 1, 2023
out = dcop(0.3,0.7,"clayton",270,2)out = dcop(0.3,0.7,"clayton",270,2)
Density at (u,v)
dfgm(u, v, cpar)dfgm(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dfgm(0.3,0.5,0.2)out = dfgm(0.3,0.5,0.2)
Density at (u,v)
dfrk(u, v, cpar)dfrk(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter, cpar>0 or cpar<0 |
out |
Vector of densities |
Pavel Krupskii
out = dfrk(0.3,0.5,2)out = dfrk(0.3,0.5,2)
Density at (u,v)
dgal(u, v, cpar)dgal(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dgal(0.3,0.5,2)out = dgal(0.3,0.5,2)
Density at (u,v)
dgum(u, v, cpar)dgum(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dgum(0.3,0.5,2)out = dgum(0.3,0.5,2)
Density at (u,v)
dhr(u, v, cpar)dhr(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dhr(0.3,0.5,2)out = dhr(0.3,0.5,2)
Density at (u,v)
djoe(u, v, cpar)djoe(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 1 |
out |
Vector of densities |
Pavel Krupskii
out = djoe(0.3,0.5,2)out = djoe(0.3,0.5,2)
Density at (u,v)
dmtcj(u, v, cpar)dmtcj(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dmtcj(0.3,0.5,2)out = dmtcj(0.3,0.5,2)
Density at (u,v)
dpla(u, v, cpar)dpla(u, v, cpar)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
cpar |
copula parameter > 0 |
out |
Vector of densities |
Pavel Krupskii
out = dpla(0.3,0.5,2)out = dpla(0.3,0.5,2)
This function computes the estimation of a copula-based 2-level hierarchical model.
EstContinuous( y, model, family, rot = 0, clu, xc = NULL, xm = NULL, start, LB, UB, nq = 31, dfM = NULL, dfC = NULL, prediction = TRUE )EstContinuous( y, model, family, rot = 0, clu, xc = NULL, xm = NULL, start, LB, UB, nq = 31, dfM = NULL, dfC = NULL, prediction = TRUE )
y |
n x 1 vector of response variable (assumed continuous). |
model |
function for margins: "gaussian" (normal), "t" (Student with known df=dfM), laplace" , "exponential", "weibull". |
family |
copula family: "gaussian" , "t" , "clayton" , "frank" , "fgm", "gumbel". |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
clu |
variable of size n defining the clusters; can be a factor |
xc |
covariates of size n for the estimation of the copula, in addition to the constant; default is NULL. |
xm |
covariates of size n for the estimation of the mean of the margin, in addition to the constant; default is NULL. |
start |
starting point for the estimation; could be the ones associated with a Gaussian-copula model defined by lmer. |
LB |
lower bound for the parameters. |
UB |
upper bound for the parameters. |
nq |
number of nodes and weighted for Gaussian quadrature of the product of conditional copulas; default is 25. |
dfM |
degrees of freedom for a Student margin; default is 0 for non-t distribution, |
dfC |
degrees of freedom for a Student margin; default is 5. |
prediction |
logical variable for prediction of latent variables V; default is TRUE. |
coefficients |
Estimated parameters |
sd |
Standard deviations of the estimated parameters |
tstat |
T statistics for the estimated parameters |
pval |
P-values of the t statistics for the estimated parameters |
gradient |
Gradient of the log-likelihood |
loglik |
Log-likelihood |
aic |
AIC coefficient |
bic |
BIC coefficient |
cov |
Covariance matrix of the estimations |
grd |
Gradients by clusters |
clu |
Cluster values |
Matxc |
Matrix of covariates defining the copula parameters, including a constant |
Matxm |
Matrix of covariates defining the margin parameters, including a constant |
V |
Estimated value of the latent variable by clusters (if prediction=TRUE) |
cluster |
Unique values of clusters |
family |
Copula family |
tau |
Kendall's tau by observation |
thC0 |
Estimated parameters of the copula by observation |
thF |
Estimated parameters of the margins by observation |
pcond |
Conditional copula cdf |
fcpdf |
Margin functions (cdf and pdf) |
dfM |
Degrees of freedom for Student margin (default is NULL) |
dfC |
Degrees of freedom for the Student copula (default is NULL) |
Pavel Krupskii, Bouchra R. Nasri and Bruno N. Remillard
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
data(normal) #simulated data with normal margins start=c(0,0,0,1); LB=c(rep(-10,3),0.001);UB=c(rep(10,3),10) y=normal$y; clu=normal$clu;xm=normal$xm out=EstContinuous(y,model="gaussian",family="clayton",rot=90,clu=clu,xm=xm,start=start,LB=LB,UB=UB)data(normal) #simulated data with normal margins start=c(0,0,0,1); LB=c(rep(-10,3),0.001);UB=c(rep(10,3),10) y=normal$y; clu=normal$clu;xm=normal$xm out=EstContinuous(y,model="gaussian",family="clayton",rot=90,clu=clu,xm=xm,start=start,LB=LB,UB=UB)
This function computes the estimation of a copula-based 2-level hierarchical model.
EstDiscrete( y, model, family, rot = 0, clu, xc = NULL, xm = NULL, start, LB, UB, nq = 25, dfC = NULL, offset = NULL, prediction = TRUE )EstDiscrete( y, model, family, rot = 0, clu, xc = NULL, xm = NULL, start, LB, UB, nq = 25, dfC = NULL, offset = NULL, prediction = TRUE )
y |
n x 1 vector of response variable (assumed continuous). |
model |
margins: "binomial" or "bernoulli","poisson", "nbinom" (Negative Binomial), "geometric", "multinomial". |
family |
copula family: "gaussian" , "t" , "clayton" , "frank" , "fgm", gumbel". |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
clu |
variable of size n defining the clusters; can be a factor |
xc |
covariates of size n for the estimation of the copula, in addition to the constant; default is NULL. |
xm |
covariates of size n for the estimation of the mean of the margin, in addition to the constant; default is NULL. |
start |
starting point for the estimation; could be the ones associated with a Gaussian-copula model defined by lmer. |
LB |
lower bound for the parameters. |
UB |
upper bound for the parameters. |
nq |
number of nodes and weighted for Gaussian quadrature of the product of conditional copulas; default is 25. |
dfC |
degrees of freedom for a Student margin; default is 0. |
offset |
offset (default is NULL) |
prediction |
logical variable for prediction of latent variables V (default is TRUE). |
coefficients |
Estimated parameters |
sd |
Standard deviations of the estimated parameters |
tstat |
T statistics for the estimated parameters |
pval |
P-values of the t statistics for the estimated parameters |
gradient |
Gradient of the log-likelihood |
loglik |
Log-likelihood |
aic |
AIC coefficient |
bic |
BIC coefficient |
cov |
Covariance matrix of the estimations |
grd |
Gradients by clusters |
clu |
Cluster values |
Matxc |
Matrix of covariates defining the copula parameters, including a constant |
Matxm |
Matrix of covariates defining the margin parameters, including a constant |
V |
Estimated value of the latent variable by clusters (if prediction=TRUE) |
cluster |
Unique clusters |
family |
Copula family |
thC0 |
Estimated parameters of the copula by observation |
thF |
Estimated parameters of the margins by observation |
rot |
rotation |
dfC |
Degrees of freedom for the Student copula |
model |
Name of the margins |
disc |
Discrete margin number |
Pavel Krupskii, Bouchra R. Nasri and Bruno N. Remillard
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
data(sim.poisson) #simulated data with Poisson margins start=c(2,8,3,-1); LB = c(-3, 3, -7, -6);UB=c( 7, 13, 13, 4) y=sim.poisson$y; clu=sim.poisson$clu; xc=sim.poisson$xc; xm=sim.poisson$xm model = "poisson"; family="frank" out.poisson=EstDiscrete(y,model,family,rot=0,clu,xc,xm,start,LB,UB,nq=31,prediction=TRUE)data(sim.poisson) #simulated data with Poisson margins start=c(2,8,3,-1); LB = c(-3, 3, -7, -6);UB=c( 7, 13, 13, 4) y=sim.poisson$y; clu=sim.poisson$clu; xc=sim.poisson$xc; xm=sim.poisson$xm model = "poisson"; family="frank" out.poisson=EstDiscrete(y,model,family,rot=0,clu,xc,xm,start,LB,UB,nq=31,prediction=TRUE)
This function computes the conditional expectation for a given copula family and a given margin variables for a clustered data model. The clusters ar3e independent but the observations with clusters are dependent, according to a one-factor copula model.
expcond(w, family, rot = 0, cpar, margin, dfC = NULL, subs = 1000)expcond(w, family, rot = 0, cpar, margin, dfC = NULL, subs = 1000)
w |
value of the conditioning random variable |
family |
copula model: "gaussian" , "t" , "clayton" ,"joe", "frank" , "gumbel", "plackett" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
cpar |
copula parameter |
margin |
marginal distribution function |
dfC |
degrees of freedom for the Student copula (default is NULL) |
subs |
number of subdivisions for the integrals (default=1000) |
mest |
Conditional expectations |
Pavel Krupskii and Bruno N. Remillard
margin = function(x){ppois(x,10)} expcond(0.4,'clayton',cpar=2,margin=margin)margin = function(x){ppois(x,10)} expcond(0.4,'clayton',cpar=2,margin=margin)
This function computes the inverse conditional expecatation for a given copula family and a given margin variables for a clustered data model. The clusters ar3e independent but the observations with clusters are dependent, according to a one-factor copula model.
expcondinv(u, family, cpar, rot = 0, margin, subs = 1000, eps = 1e-04)expcondinv(u, family, cpar, rot = 0, margin, subs = 1000, eps = 1e-04)
u |
conditional expectation |
family |
copula model: "gaussian" , "t" , "clayton" "joe", "frank" , "gumbel", "plackett" |
cpar |
copula parameter |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
margin |
marginal distribution function of the response |
subs |
number of subdivisions for the integrals (default=1000) |
eps |
precision required |
minv |
Inverse conditional expectation |
Pavel Krupskii and Bruno N. Remillard
Inverse conditional expectation for a single value
expcondinv1(u, family, cpar, rot = 0, margin, subs = 1000, eps = 1e-04)expcondinv1(u, family, cpar, rot = 0, margin, subs = 1000, eps = 1e-04)
u |
conditional expectation |
family |
copula model: "gaussian" , "t" , "clayton" "joe", "frank" , "gumbel", "plackett" |
cpar |
copula parameter |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
margin |
marginal distribution function of the response |
subs |
number of subdivisions for the integrals (default=1000) |
eps |
precision required |
minv |
Inverse conditional expectation |
This function computes the cdf, pdf, and associated derivatives
expcpdf(z, th)expcpdf(z, th)
z |
vector of responses |
th |
th is rate > 0 |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = expcpdf(2,3)out = expcpdf(2,3)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
ffgmders(u, v, cpar, du = FALSE)ffgmders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter in [-1,1] |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = ffgmders(0.3,0.5,2,TRUE)out = ffgmders(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
ffrkders(u, v, cpar, du = FALSE)ffrkders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = ffrkders(0.3,0.5,2,TRUE)out = ffrkders(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
fgumders(u, v, cpar, du = FALSE)fgumders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter > 1 |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = fgumders(0.3,0.5,2,TRUE)out = fgumders(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
fjoeders(u, v, cpar, du = FALSE)fjoeders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter > 1 |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = fjoeders(0.3,0.5,2,TRUE)out = fjoeders(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
fmtcjders(u, v, cpar, du = FALSE)fmtcjders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter > 0 |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = fmtcjders(0.3,0.5,2,TRUE)out = fmtcjders(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
fnorders(u, v, cpar, du = FALSE)fnorders(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter in (-1,1) |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = fnorders(0.3,0.5,0.6,TRUE)out = fnorders(0.3,0.5,0.6,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
fpladers(u, v, cpar, du = FALSE)fpladers(u, v, cpar, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter > 0 |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = fpladers(0.3,0.5,2,TRUE)out = fpladers(0.3,0.5,2,TRUE)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
ftders(u, v, cpar, nu, du = FALSE)ftders(u, v, cpar, nu, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter in (-1,1) |
nu |
degrees of freedom >0 |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = ftders(0.3,0.5,2,25)out = ftders(0.3,0.5,2,25)
Derivatives C(u|v), C'_dl(u|v), c(u,v), c'_dl(u,v), c'_u(u,v) for the linking copula
ftdersP(u, v, cpar, dfC, du = FALSE)ftdersP(u, v, cpar, dfC, du = FALSE)
u |
vector of values in (0,1) |
v |
conditioning variable in (0,1) |
cpar |
copula parameter in (-1,1) |
dfC |
degrees of freedom |
du |
logical value (default = FALSE) for the derivative of the copula density with respect to u |
out |
Matrix of conditional cdf, pdf, and derivatives with respect to parameter |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = ftdersP(0.3,0.5,2,25,TRUE)out = ftdersP(0.3,0.5,2,25,TRUE)
This function computes the cdf, pdf, and associated derivatives
geomcpdf(z, th)geomcpdf(z, th)
z |
vector of responses |
th |
linear combination of covariates (can be negative) |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = geomcpdf(0,-3)out = geomcpdf(0,-3)
This function is used to get the inverse of a monotonic function on (0,1), depending on parameters, and using the bisection method
invfunc(q, func, th, lb = 1e-12, ub = 1 - 1e-12, tol = 1e-08, nbreak = 40)invfunc(q, func, th, lb = 1e-12, ub = 1 - 1e-12, tol = 1e-08, nbreak = 40)
q |
Function value (can be a vector if func() supports a vector argument) |
func |
Function of one argument to be inverted |
th |
Function parameters |
lb |
Lower bound for the possible values |
ub |
Upper bound for the possible values |
tol |
Tolerance for the inversion |
nbreak |
Maximum number of iterations (default is 40) |
out |
Inverse values |
Pavel Krupskii
This function computes the cdf, pdf, and associated derivatives
lapcpdf(z, th)lapcpdf(z, th)
z |
vector of responses |
th |
th[,1] is mean, th[,2] is standard deviation > 0 |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = lapcpdf(2,c(-3,4))out = lapcpdf(2,c(-3,4))
Computes the copula parameters given a linear combination of covariates.
linkCop(th, family = "clayton")linkCop(th, family = "clayton")
th |
vector of linear combinations |
family |
copula family: "gaussian" , "t" , "clayton" , "claytonR" , "frank" , "gumbel", "gumbelR". |
cpar |
Associated copula parameters |
hder |
Derivative of link function |
Pavel Krupskii and Bruno N. Remillard, January 20, 2023
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
out = linkCop(-1,"gaussian")out = linkCop(-1,"gaussian")
This function computes the estimation of a latent variables for each cluster using the conditional a posteriori median.
MAP.continuous(u, family, rot, thC0k, dfC = NULL, nq = 35)MAP.continuous(u, family, rot, thC0k, dfC = NULL, nq = 35)
u |
vector of values in (0,1) |
family |
copula family: "gaussian" , "t" , "clayton" , "joe", "frank" , "fgm", gumbel", "plackett", "galambos", "huesler-reiss" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270. |
thC0k |
vector of copula parameters |
dfC |
degrees of freedom for the Student copula (default is NULL) |
nq |
number of nodes and weighted for Gaussian quadrature of the product of conditional copulas; default is 31. |
condmed |
Conditional a posteriori median. |
Pavel Krupskii, Bouchra R. Nasri and Bruno N. Remillard
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
u = c(0.5228155, 0.3064417, 0.2789849, 0.5176489, 0.3587144) thC0k=rep(17.54873,5) MAP.continuous(u,"clayton",rot=90,thC0k,nq=35)u = c(0.5228155, 0.3064417, 0.2789849, 0.5176489, 0.3587144) thC0k=rep(17.54873,5) MAP.continuous(u,"clayton",rot=90,thC0k,nq=35)
This function computes the estimation of a latent variables foe=r each cluster using the conditional a posteriori median.
MAP.discrete(vv, uu, family, rot, thC0k, dfC = NULL, nq = 35)MAP.discrete(vv, uu, family, rot, thC0k, dfC = NULL, nq = 35)
vv |
vector of values in (0,1) |
uu |
vector of values in (0,1) |
family |
copula family "gaussian" , "t" , "clayton" , "joe", "frank" , "fgm", gumbel", "plackett", "galambos", "huesler-reiss" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270. |
thC0k |
vector of copula parameters |
dfC |
degrees of freedom for the Student copula (default is NULL) |
nq |
number of nodes and weighted for Gaussian quadrature of the product of conditional copulas; default is 31. |
condmed |
Conditional a posteriori median. |
Pavel Krupskii, Bouchra R. Nasri and Bruno N. Remillard
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
uu = c(0.5228155, 0.3064417, 0.2789849, 0.5176489, 0.3587144) vv = c(0.7816627, 0.6688788, 0.6351364, 0.7774917, 0.7264787) thC0k=rep(17.54873,5) MAP.discrete(vv,uu,"clayton",rot=90,thC0k,nq=35)uu = c(0.5228155, 0.3064417, 0.2789849, 0.5176489, 0.3587144) vv = c(0.7816627, 0.6688788, 0.6351364, 0.7774917, 0.7264787) thC0k=rep(17.54873,5) MAP.discrete(vv,uu,"clayton",rot=90,thC0k,nq=35)
This function computes the cdf, pdf, and associated derivatives
margins(z, th, model, x = NULL, dfM = NULL)margins(z, th, model, x = NULL, dfM = NULL)
z |
vector of responses |
th |
linear combination of covariates (can be negative) |
model |
model for margin: "binomial" (bernoulli), "poisson", "nbinom" (mean is the parameter),"nbinom1" (p is the parameter), "geometric", "multinomial", "exponential", "weibull", "normal","t", "laplace" |
x |
covariates for the multinomial margin (default is NULL) |
dfM |
degrees of freedom for the Student margin (default is NULL) |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = margins(0,2.5,"binomial")out = margins(0,2.5,"binomial")
Computes the MLE estimation for a bivariate copula using gradient. The likelihood is likelihood is c(u,v;theta)
mlecop(u, v, fcopders, start = 2, LB = 1.01, UB = 7)mlecop(u, v, fcopders, start = 2, LB = 1.01, UB = 7)
u |
vector of values in (0,1) |
v |
vector of values in (0,1) |
fcopders |
ffrkders, fgumders or fmtcjders |
start |
starting value for the parameter (default =2) |
LB |
lower bound for the parameter (default is 1.01) |
UB |
upper bound for the parameter (default is 7) |
mle |
List of outputs from nlm function |
Pavel Krupskii
set.seed(2) v = runif(250) w = runif(250) u = 1/sqrt(1+(w^(-2/3)-1)/v^2) # Clayton copula with parameter 2 (tau=0.5) out = mlecop(u,v,fmtcjders)set.seed(2) v = runif(250) w = runif(250) u = 1/sqrt(1+(w^(-2/3)-1)/v^2) # Clayton copula with parameter 2 (tau=0.5) out = mlecop(u,v,fmtcjders)
Computes the MLE estimation for a bivariate copula using gradient. The likelihood is likelihood is C(1-p|v;theta) if y=0 and 1-C(1-p|v;theta) if y=1
mlecop.disc(y, v, fcopders, start = 2, LB = 1.01, UB = 7)mlecop.disc(y, v, fcopders, start = 2, LB = 1.01, UB = 7)
y |
vector of binary values 0 or 1 |
v |
vector of values in (0,1) |
fcopders |
ffrkders, fgumders or fmtcjders |
start |
starting value for the parameter (default =2) |
LB |
lower bound for the parameter (default is 1.01) |
UB |
upper bound for the parameter (default is 7) |
mle |
List of outputs from nlm function |
Pavel Krupskii
set.seed(2) v = runif(250) w = runif(250) u = 1/sqrt(1+(w^(-2/3)-1)/v^2) #Clayton with parameter 2 y = as.numeric(u>0.6) # if one takes (u<4), one obtains a rotation of the Clayton! out = mlecop.disc(y,v,fmtcjders)set.seed(2) v = runif(250) w = runif(250) u = 1/sqrt(1+(w^(-2/3)-1)/v^2) #Clayton with parameter 2 y = as.numeric(u>0.6) # if one takes (u<4), one obtains a rotation of the Clayton! out = mlecop.disc(y,v,fmtcjders)
This function computes the cdf, pdf, and associated derivatives
multinomcpdf(z, th, x)multinomcpdf(z, th, x)
z |
vector of responses taking values in 1,...,nL: as.number(z) if z is a factor! |
th |
th is a n x (L-1) matrix of parameters, i.e., mpar = a=[a_1,1,...a_1,k2,a_2,1,...a_2,k2,... a_L-1,1... a_L-1,k2], and first level is the baseline. |
x |
matrix of covariates (including the constant) |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
x=matrix(c(1,1,-1,-1,0,2),nrow=2) z = c(1,3) th = matrix(c(1,2,3,4,5,6),nrow=2) out = multinomcpdf(z,th,x = x)x=matrix(c(1,1,-1,-1,0,2),nrow=2) z = c(1,3) th = matrix(c(1,2,3,4,5,6),nrow=2) out = multinomcpdf(z,th,x = x)
Simulated clustered data from a Clayton copula with parameter 2, and multinomial margins with 3 levels and parameters 1.0,-1 for level 2 and 0.5, 2 for level 3. Clusters and covariates are included.
data(multinomial)data(multinomial)
Data frame of numerical values
data(multinomial)data(multinomial)
This function computes the cdf, pdf, and associated derivatives
nbinom1cpdf(z, th)nbinom1cpdf(z, th)
z |
vector of responses |
th |
th[,1] is size > 0 and th[,2] is mean > 0; size does not have to be integer |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = nbinom1cpdf(0,c(1,0.5))out = nbinom1cpdf(0,c(1,0.5))
This function computes the cdf, pdf, and associated derivatives
nbinomcpdf(z, th)nbinomcpdf(z, th)
z |
vector of responses |
th |
th[,1] is size > 0 and th[,2] is p, with 0<p<1; size does not have to be integer |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = nbinomcpdf(0,c(1,0.5))out = nbinomcpdf(0,c(1,0.5))
Simulated clustered data from a Clayton copula with parameter 2, rotation = 90, and normal margins with 1,-1 for the mean, ans sd = 4. Clusters and covariates are included.
data(normal)data(normal)
List of simulated values (y, clu, xm)
data(normal)data(normal)
This function computes the cdf, pdf, and associated derivatives
normcpdf(z, th)normcpdf(z, th)
z |
vector of responses |
th |
th[,1] is mean, th[,2] is standard deviation > 0; |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = normcpdf(2,c(-3,4))out = normcpdf(2,c(-3,4))
Output of EstContinuous for the simulated clustered data normal.
data(out.normal)data(out.normal)
Data frame of numerical values
data(out.normal)data(out.normal)
Output of EstDiscrete for the simulated clustered data poisson.
data(out.poisson)data(out.poisson)
Data frame of numerical values
data(out.poisson)data(out.poisson)
This function computes the conditional cdf C(U|V) for a copula C
pcond(U, V, family, rot = 0, cpar, dfC = NULL)pcond(U, V, family, rot = 0, cpar, dfC = NULL)
U |
values at which the cdf is evaluated |
V |
value of the conditioning variable in (0,1) |
family |
"gaussian" , "t" , "clayton" , "joe", "frank" , "fgm", gumbel", "plackett", "galambos", "huesler-reiss" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
cpar |
copula parameter (vector) |
dfC |
degrees of freedom of the Student copula (default is NULL) |
p |
Conditional cdf |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
p = pcond(0.1,0.2,"clayton",rot=270,cpar=0.87)p = pcond(0.1,0.2,"clayton",rot=270,cpar=0.87)
Conditional Clayton
pcondcla(u, v, cpar)pcondcla(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
ccdf |
Conditional cdf |
pcondcla(0.5,0.6,2)pcondcla(0.5,0.6,2)
Conditional FGM (B10)
pcondfgm(u, v, cpar)pcondfgm(u, v, cpar)
u |
probability |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter -1<=cpar<=1 |
ccdf |
Conditional cdf |
pcondfgm(0.5,0.6,0.9)pcondfgm(0.5,0.6,0.9)
Conditional Frank (B3)
pcondfrk(u, v, cpar)pcondfrk(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
ccdf |
Conditional cdf |
pcondfrk(0.5,0.6,2)pcondfrk(0.5,0.6,2)
Conditional Galambos (B7)
pcondgal(u, v, cpar)pcondgal(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
ccdf |
Conditional cdf |
pcondgal(0.5,0.6,2)pcondgal(0.5,0.6,2)
Conditional Gumbel (B6)
pcondgum(u, v, cpar)pcondgum(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter >1 |
ccdf |
Conditional cdf |
pcondgum(0.5,0.6,2)pcondgum(0.5,0.6,2)
Conditional Huesler-Reiss (B8)
pcondhr(u, v, cpar)pcondhr(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter >0 |
ccdf |
Conditional cdf |
pcondhr(0.5,0.6,2)pcondhr(0.5,0.6,2)
Conditional Joe (B5)
pcondjoe(u, v, cpar)pcondjoe(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
ccdf |
Conditional cdf |
pcondjoe(0.5,0.6,2)pcondjoe(0.5,0.6,2)
Conditional Gaussian
pcondnor(u, v, cpar)pcondnor(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
ccdf |
Conditional cdf |
pcondnor(0.5,0.6,0.6)pcondnor(0.5,0.6,0.6)
Conditional Plackett (B2)
pcondpla(u, v, cpar)pcondpla(u, v, cpar)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter >1 |
ccdf |
Conditional cdf |
pcondpla(0.5,0.6,2)pcondpla(0.5,0.6,2)
Conditional Student is Y2|Y1=y1 ~ t(nu+1,location=rho*y1, sigma(y1)), where here sigma^2 = (1-rho^2)(nu+y1^2)/(nu+1)
pcondt(u, v, cpar, dfC)pcondt(u, v, cpar, dfC)
u |
values at which the cdf is evaluated |
v |
value of the conditioning variable in (0,1) |
cpar |
copula parameter |
dfC |
degrees of freedom |
ccdf |
Conditional cdf |
pcondt(0.5,0.6,0.6,15)pcondt(0.5,0.6,0.6,15)
This function computes the cdf, pdf, and associated derivatives
poiscpdf(z, th)poiscpdf(z, th)
z |
vector of responses |
th |
values of lambda >0 |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = poiscpdf(0,2.5)out = poiscpdf(0,2.5)
Compute the conditional expectation of a copula-based 2-level hierarchical model for continuous response.
predictContinuous(object, newdata = NULL, nq = 25)predictContinuous(object, newdata = NULL, nq = 25)
object |
Object of class “EstContinuous“ generated by EstContinuous. |
newdata |
List of variables for be predicted (“clu“ for clusters, “xc“ for the copula covariates, and “xm“ for the margins covariates). The covariates can be NULL. |
nq |
number of nodes and weighted for Gaussian quadrature of the product of conditional copulas; default is 25. |
mest |
Conditional expectations |
Pavel Krupskii and Bruno N. Remillard, January 20, 2023
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
data(out.normal) newdata=list(clu=c(1:50),xm=rep(0.4,50)) pred= predictContinuous(out.normal,newdata)data(out.normal) newdata=list(clu=c(1:50),xm=rep(0.4,50)) pred= predictContinuous(out.normal,newdata)
Compute the conditional expectation of a copula-based 2-level hierarchical model for disctrete response.
predictDiscrete(object, newdata, m = 100)predictDiscrete(object, newdata, m = 100)
object |
Object of class “EstDiscrete“ generated by EstDiscrete. |
newdata |
List of variables for be predicted (“clu“ for clusters, “xc“ for the copula covariates, and “xm“ for the margins covariates). The covariates can be NULL. |
m |
Number of points for the numerical integration (default is 100). |
mest |
Conditional expectations (conditional probabilities for the multinomial case |
Pavel Krupskii and Bruno N. Remillard, January 20, 2023
Krupskii, Nasri & Remillard (2023). On factor copula-based mixed regression models
data(out.poisson) newdata = list(clu=c(1:50),xc=rep(0.2,50),xm=rep(0.5,50)) pred= predictDiscrete(out.poisson,newdata,m=100)data(out.poisson) newdata = list(clu=c(1:50),xc=rep(0.2,50),xm=rep(0.5,50)) pred= predictDiscrete(out.poisson,newdata,m=100)
This function estimates the empirical cdf, its left limit, and pseudo-observations for a univatiate vector y
pseudosC(y)pseudosC(y)
y |
univariate data |
Fn |
Emprirical cdf |
Fm |
Left-contniuous cdf |
U |
Pseudo-obsevations |
Bruno N. Remillard, January 20, 2022
y = rpois(100,2) out=pseudosC(y)y = rpois(100,2) out=pseudosC(y)
This function computes the quantile of conditional cdf C(U|v) for a copula C
qcond(w, v, family, cpar, rot = 0)qcond(w, v, family, cpar, rot = 0)
w |
probability |
v |
value of the conditioning variable in (0,1) |
family |
"gaussian" , "t" , "clayton" , "fgm", "frank" , "gumbel", "plackett", "galambos", "huesler-reiss" |
cpar |
copula parameter (vector) |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
U |
Conditional quantile |
U |
Conditional quantile |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
U = qcond(0.1,0.2,"gaussian",0.87)U = qcond(0.1,0.2,"gaussian",0.87)
Inverse clayton
qcondcla(w, v, th)qcondcla(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter |
out |
Conditional quantile |
Inverse FGM (B10)
qcondfgm(w, v, th)qcondfgm(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter -1<=th<=1 |
out |
Conditional quantile |
Inverse Frank
qcondfra(w, v, th)qcondfra(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter |
out |
Conditional quantile |
Inverse Galambos
qcondgal(w, v, th)qcondgal(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter >0 |
out |
Conditional quantile |
Inverse Gumbel
qcondgum(w, v, th)qcondgum(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter |
out |
Conditional quantile |
Inverse Huesler-Reiss
qcondhr(w, v, th)qcondhr(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter >0 |
out |
Conditional quantile |
Inverse Joe
qcondjoe(w, v, th)qcondjoe(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter >-1 |
out |
Conditional quantile |
Inverse Gaussian
qcondnor(w, v, th)qcondnor(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter (correlation) |
out |
Conditional quantile |
Inverse Plackett
qcondpla(w, v, th)qcondpla(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter |
out |
Conditional quantile |
Inverse Student
qcondt(w, v, th)qcondt(w, v, th)
w |
probability |
v |
value of the conditioning variable in (0,1) |
th |
copula parameter |
out |
Conditional quantile |
Simulated clustered data from a Frank copula with parC=c(2,8), and Poisson margins with parM=c(3.0,-0.1). Clusters and covariates (both uniform) are included.
data(sim.poisson)data(sim.poisson)
List of simulated values (y, clu,xc,xm) together with true parameters
data(sim.poisson)data(sim.poisson)
Generate a random sample of observations from a copula-based mixed regression model.
SimGenCluster( parC, parM, clu, xc = NULL, xm = NULL, family, rot = 0, dfC = NULL, model, dfM = NULL, offset = NULL )SimGenCluster( parC, parM, clu, xc = NULL, xm = NULL, family, rot = 0, dfC = NULL, model, dfM = NULL, offset = NULL )
parC |
vector of copula parameters; k1 is the number of covariates + constant for the copula |
parM |
vector of margin parameters; k2 is the number of covariates + constant for the margins |
clu |
vector of clusters (can be a factor) |
xc |
matrix (N x k1) of covariates for the copula, not including the constant (can be NULL) |
xm |
matrix (N x k2) of covariates for the margins, not including the constant (can be NULL) |
family |
copula family: "gaussian" , "t" , "clayton" , "joe", "frank" , "gumbel", "plackett" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
dfC |
degrees of freedom for the Student copula (default is NULL) |
model |
marginal distribution: "binomial" (bernoulli), "poisson", "nbinom" (mean is the parameter),"nbinom1" (p is the parameter), "geometric", "multinomial", exponential", "weibull", "normal" (gaussian),"t", "laplace" |
dfM |
degrees of freedom for the Student margins (default is NULL) |
offset |
offset for the margins (default is NULL) |
y |
Simulated response |
y |
Simulated values |
Bruno N. Remillard
K=50 #number of clusters n=5 #size of each cluster N=n*K set.seed(1) clu=rep(c(1:K),each=n) parC = 0 # yields tau = 0.5 for Clayton parM= c(1,-1,4) xm = runif(N) y=SimGenCluster(parC,parM,xm,family="clayton",rot=90,clu=clu,model="gaussian")K=50 #number of clusters n=5 #size of each cluster N=n*K set.seed(1) clu=rep(c(1:K),each=n) parC = 0 # yields tau = 0.5 for Clayton parM= c(1,-1,4) xm = runif(N) y=SimGenCluster(parC,parM,xm,family="clayton",rot=90,clu=clu,model="gaussian")
Generate a random sample of multinomial observations from a copula-based mixed regression model.
SimMultinomial( parC, parM, clu, xc = NULL, xm = NULL, family, rot = 0, dfC = NULL, offset = NULL )SimMultinomial( parC, parM, clu, xc = NULL, xm = NULL, family, rot = 0, dfC = NULL, offset = NULL )
parC |
copula parameters |
parM |
matrix of dimension (L-1)x k2 of margin parameters; L is the number of levels and k2 is the number of covariates+constant for the margins |
clu |
vector of clusters (can be a factor) |
xc |
matrix of covariates for the copula, not including the constant (can be NULL) |
xm |
matrix of covariates for the margins, not including the constant (can be NULL) |
family |
copula family: "gaussian" , "t" , "clayton" , "joe", "frank" , "gumbel", "plackett" |
rot |
rotation: 0 (default), 90, 180 (survival), or 270 |
dfC |
degrees of freedom for student copula (default is NULL) |
offset |
offset for the margins (default is NULL) |
y |
Simulated factor |
Bruno N. Remillard
K=50 #number of clusters n=5 #size of each cluster N=n*K set.seed(1) clu=rep(c(1:K),each=n) parC = 2 parM=matrix(c(1,-1,0.5,2),byrow=TRUE,ncol=2) xm = runif(N) y=SimMultinomial(parC,parM,clu,xm=xm,family="clayton",rot=90)K=50 #number of clusters n=5 #size of each cluster N=n*K set.seed(1) clu=rep(c(1:K),each=n) parC = 2 parM=matrix(c(1,-1,0.5,2),byrow=TRUE,ncol=2) xm = runif(N) y=SimMultinomial(parC,parM,clu,xm=xm,family="clayton",rot=90)
This function computes the cdf, pdf, and associated derivatives
tcpdf(z, th, df)tcpdf(z, th, df)
z |
vector of responses |
th |
th[,1] is mean, th[,2] is standard deviation > 0 |
df |
degrees of freedom |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = tcpdf(2,c(-3,4),25)out = tcpdf(2,c(-3,4),25)
This function computes the cdf, pdf, and associated derivatives
weibcpdf(z, th)weibcpdf(z, th)
z |
vector of responses |
th |
th[,1] is rate>0, th[,2] is shape > 0; |
out |
Matrix of conditional cdf, derivative with respect to parameter, pdf, |
Pavel Krupskii and Bruno N. Remillard, January 20, 2022
out = weibcpdf(2,c(2,3))out = weibcpdf(2,c(2,3))