R/ST3.R

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

# updated 27_11_2007
ST3 <- function (mu.link="identity", sigma.link="log", nu.link ="log", tau.link="log")
{
  mstats <- checklink("mu.link", "skew t type 2", substitute(mu.link), 
                      c("inverse", "log", "identity", "own"))
  dstats <- checklink("sigma.link", "skew t type 2", substitute(sigma.link), 
                      c("inverse", "log", "identity", "own"))
  vstats <- checklink("nu.link", "skew t type 2",substitute(nu.link), 
                      c("inverse", "log", "identity", "own"))
  tstats <- checklink("tau.link", "skew t type 2 ",substitute(tau.link), 
                      c("inverse", "log", "identity", "own")) 
  structure(
    list(family = c("ST3",  "skew t type 3"),
         parameters = list(mu=TRUE, sigma=TRUE, nu=TRUE, tau=TRUE), 
         nopar = 4, 
         type = "Continuous",
         mu.link = as.character(substitute(mu.link)),  
         sigma.link = as.character(substitute(sigma.link)), 
         nu.link = as.character(substitute(nu.link)), 
         tau.link = as.character(substitute(tau.link)), 
         mu.linkfun = mstats$linkfun, 
         sigma.linkfun = dstats$linkfun, 
         nu.linkfun = vstats$linkfun,
         tau.linkfun = tstats$linkfun,  
         mu.linkinv = mstats$linkinv, 
         sigma.linkinv = dstats$linkinv,
         nu.linkinv = vstats$linkinv,
         tau.linkinv = tstats$linkinv, 
         mu.dr = mstats$mu.eta, 
         sigma.dr = dstats$mu.eta, 
         nu.dr = vstats$mu.eta,
         tau.dr = tstats$mu.eta, 
         dldm = function(y,mu,sigma,nu,tau) {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)/s1)^2
           dsq2 <- ((y-mu)/s2)^2
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldm <- ifelse(y < mu, (w1*(y-mu))/(s1^2) , (w2*(y-mu))/(s2^2))
           dldm
         },
         d2ldm2 = function(y,mu,sigma,nu,tau){
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <-((y-mu)^2)/(s2^2) 
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldm <- ifelse(y < mu, (w1*(y-mu))/(s1^2) , (w2*(y-mu))/(s2^2) )
           d2ldm2 <- -dldm*dldm
           d2ldm2 <- ifelse(d2ldm2 < -1e-15, d2ldm2,-1e-15)  # NOTE this was added afrer testing
           d2ldm2
         },
         dldd = function(y,mu,sigma,nu,tau) {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldd <- ifelse(y < mu, (w1*dsq1-1)/(sigma) , (w2*dsq2-1)/(sigma) )
           dldd
         } ,
         d2ldd2 = function(y,mu,sigma,nu,tau) {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldd <- ifelse(y < mu, (w1*dsq1-1)/(sigma) , (w2*dsq2-1)/(sigma) )
           d2ldd2 <- -dldd*dldd
           d2ldd2 <- ifelse(d2ldd2 < -1e-15, d2ldd2,-1e-15)  
           d2ldd2
         },
         dldv = function(y,mu,sigma,nu,tau) {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldv <- ifelse(y < mu, -(w1*dsq1-1)/(nu) , (w2*dsq2+1)/(nu) )
           dldv <- dldv - 2*nu/(1+nu^2)
           dldv
         } ,
         d2ldv2 = function(y,mu,sigma,nu,tau) { 
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldv <- ifelse(y < mu, -(w1*dsq1-1)/(nu) , (w2*dsq2+1)/(nu) )
           dldv <- dldv - 2*nu/(1+nu^2)
           d2ldv2 <- -dldv*dldv
           d2ldv2 <- ifelse(d2ldv2 < -1e-15, d2ldv2,-1e-15)  
           d2ldv2
           
         },
         dldt = function(y,mu,sigma,nu,tau) { 
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldta <- -0.5*log(1+dsq1/tau)+(w1*dsq1-1)/(2*tau)
           dldtb <- -0.5*log(1+dsq2/tau)+(w2*dsq2-1)/(2*tau)
           dldt <- ifelse(y < mu, dldta , dldtb)
           dldt <- dldt+0.5*digamma((tau+1)/2)-0.5*digamma(tau/2)
           dldt
         } ,
         d2ldt2 = function(y,mu,sigma,nu,tau) {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldta <- -0.5*log(1+dsq1/tau)+(w1*dsq1-1)/(2*tau)
           dldtb <- -0.5*log(1+dsq2/tau)+(w2*dsq2-1)/(2*tau)
           dldt <- ifelse(y < mu, dldta , dldtb)
           dldt <- dldt+0.5*digamma((tau+1)/2)-0.5*digamma(tau/2)
           d2ldt2 <- -dldt*dldt
           d2ldt2 <- ifelse(d2ldt2 < -1e-15, d2ldt2,-1e-15)                                    
           d2ldt2
         } ,
         d2ldmdd = function(y,mu,sigma,nu,tau)  
         {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)/s1)^2
           dsq2 <- ((y-mu)/s2)^2
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldm <- ifelse(y < mu, (w1*(y-mu))/(s1^2) , (w2*(y-mu))/(s2^2))
           dldd <- ifelse(y < mu, (w1*dsq1-1)/(sigma) , (w2*dsq2-1)/(sigma) )                
           d2ldmdd <- -(dldm*dldd)
           d2ldmdd                  
         },
         d2ldmdv = function(y,mu,sigma,nu,tau)
         {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)/s1)^2
           dsq2 <- ((y-mu)/s2)^2
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldm <- ifelse(y < mu, (w1*(y-mu))/(s1^2) , (w2*(y-mu))/(s2^2))
           dldv <- ifelse(y < mu, -(w1*dsq1-1)/(nu) , (w2*dsq2+1)/(nu) )
           dldv <- dldv - 2*nu/(1+nu^2)                       
           d2ldmdv <- -(dldm*dldv)
           d2ldmdv
         } ,
         d2ldmdt = function(y,mu,sigma,nu,tau)
         {  
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)/s1)^2
           dsq2 <- ((y-mu)/s2)^2
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldm <- ifelse(y < mu, (w1*(y-mu))/(s1^2) , (w2*(y-mu))/(s2^2))                      
           dldta <- -0.5*log(1+dsq1/tau)+(w1*dsq1-1)/(2*tau)
           dldtb <- -0.5*log(1+dsq2/tau)+(w2*dsq2-1)/(2*tau)
           dldt <- ifelse(y < mu, dldta , dldtb)
           dldt <- dldt+0.5*digamma((tau+1)/2)-0.5*digamma(tau/2)                   
           d2ldmdt <- -(dldm*dldt)
           d2ldmdt
         },
         d2ldddv = function(y,mu,sigma,nu,tau)  
         {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldd <- ifelse(y < mu, (w1*dsq1-1)/(sigma) , (w2*dsq2-1)/(sigma) )
           dldv <- ifelse(y < mu, -(w1*dsq1-1)/(nu) , (w2*dsq2+1)/(nu) )
           dldv <- dldv - 2*nu/(1+nu^2)         
           d2ldddv <- -(dldd*dldv)
           d2ldddv
         },
         d2ldddt = function(y,mu,sigma,nu,tau)  
         {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldd <- ifelse(y < mu, (w1*dsq1-1)/(sigma) , (w2*dsq2-1)/(sigma) )      
           dldta <- -0.5*log(1+dsq1/tau)+(w1*dsq1-1)/(2*tau)
           dldtb <- -0.5*log(1+dsq2/tau)+(w2*dsq2-1)/(2*tau)
           dldt <- ifelse(y < mu, dldta , dldtb)
           dldt <- dldt+0.5*digamma((tau+1)/2)-0.5*digamma(tau/2)            
           d2ldddt <- -(dldd*dldt) 
           d2ldddt                  
         },
         d2ldvdt = function(y,mu,sigma,nu,tau)  
         {
           s1 <- sigma/nu
           s2 <- sigma*nu
           dsq1 <- ((y-mu)^2)/(s1^2)
           dsq2 <- ((y-mu)^2)/(s2^2)
           w1 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq1),1)
           w2 <- ifelse(tau < 1000000, (tau+1)/(tau+dsq2),1)
           dldv <- ifelse(y < mu, -(w1*dsq1-1)/(nu) , (w2*dsq2+1)/(nu) )
           dldv <- dldv - 2*nu/(1+nu^2)
           dldta <- -0.5*log(1+dsq1/tau)+(w1*dsq1-1)/(2*tau)
           dldtb <- -0.5*log(1+dsq2/tau)+(w2*dsq2-1)/(2*tau)
           dldt <- ifelse(y < mu, dldta , dldtb)
           dldt <- dldt+0.5*digamma((tau+1)/2)-0.5*digamma(tau/2)          
           d2ldvdt <- -(dldv*dldt)
           d2ldvdt
           
         }, 
         G.dev.incr  = function(y,mu,sigma,nu,tau,...) 
           -2*dST3(y,mu,sigma,nu,tau,log=TRUE), 
         rqres = expression(rqres(pfun="pST3", type="Continuous", y=y, mu=mu, sigma=sigma, nu=nu, tau=tau)) ,
         mu.initial = expression(mu <- (y+mean(y))/2), 
         sigma.initial = expression(sigma<- rep(sd(y), length(y))),
         nu.initial = expression(nu <- rep(1, length(y))), 
         tau.initial = expression(tau <-rep(10, length(y))), 
         mu.valid = function(mu) TRUE, 
         sigma.valid = function(sigma)  all(sigma > 0),
         nu.valid = function(nu) all(nu > 0), 
         tau.valid = function(tau) all(tau > 0), 
         y.valid = function(y)  TRUE,
         mean = function(mu, sigma, nu, tau) {
                         if (tau > 1) {
                           EZ <- (2 * sqrt(tau) * (nu - 1/nu)) / ( (tau - 1) * beta(1/2,tau/2))                     
                           return(mu + sigma * EZ)
                         } else {
                           return(NaN)
                         }
                       },
         variance = function(mu, sigma, nu, tau) {
                             if (tau > 2) {
                               EZ <- (2 * sqrt(tau) * (nu - 1/nu)) / ( (tau - 1) * beta(1/2,tau/2))  
                               return( sigma^2 * ( tau / (tau-2) * (nu^2 + nu^(-2) - 1) - (EZ)^2 ))
                             } else {
                               return(NaN)
                             }
                           }
    ),
    class = c("gamlss.family","family"))
}
#-----------------------------------------------------------------  
dST3 <- function(x, mu=0, sigma=1, nu=1, tau=10, log=FALSE)
{
  if (any(sigma <= 0))  stop(paste("sigma must be positive", "\n", "")) 
  if (any(nu <= 0))  stop(paste("nu must be positive", "\n", ""))
  if (any(tau <= 0))  stop(paste("tau must be positive", "\n", ""))  
  loglik1a <- dt((nu*(x-mu)/sigma), df=tau, log=TRUE)
  loglik2a <- dt((x-mu)/(sigma*nu), df=tau, log=TRUE)
  loglika <- ifelse(x < mu, loglik1a, loglik2a)
  loglika <- loglika+log(2*nu/(1+nu^2)) - log(sigma)
  loglik1b <- dNO((nu*(x-mu)/sigma), mu=0, sigma=1, log=TRUE)
  loglik2b <- dNO((x-mu)/(sigma*nu), mu=0, sigma=1, log=TRUE)
  loglikb <- ifelse(x < mu, loglik1b, loglik2b)
  loglikb <- loglikb+log(2*nu/(1+nu^2)) - log(sigma)
  if (length(tau)>1) loglik <- ifelse(tau<1000000, 
                                      loglika, 
                                      loglikb) 
  else loglik <- if (tau<1000000) loglika
  else  loglikb
  if(log==FALSE) ft  <- exp(loglik) else ft <- loglik 
  ft
}    
#-----------------------------------------------------------------  
pST3 <- function(q, mu=0, sigma=1, nu=1, tau=10, lower.tail = TRUE, log.p = FALSE)
{  
  if (any(sigma < 0))  stop(paste("sigma must be positive", "\n", "")) 
  if (any(nu <= 0))  stop(paste("nu must be positive", "\n", ""))
  if (any(tau < 0))  stop(paste("tau must be positive", "\n", ""))  
  cdf1 <- 2*pt(nu*(q-mu)/sigma, df=tau)
  cdf2 <- 1 + 2*nu*nu*(pt((q-mu)/(sigma*nu), df=tau) - 0.5)
  cdf <- ifelse(q < mu, cdf1, cdf2)
  cdf <- cdf/(1+nu^2)
  if(lower.tail==TRUE) cdf  <- cdf else  cdf <- 1-cdf 
  if(log.p==FALSE) cdf  <- cdf else  cdf <- log(cdf) 
  cdf    
}
#-----------------------------------------------------------------  
qST3 <- function(p, mu=0, sigma=1, nu=1, tau=10, lower.tail = TRUE, log.p = FALSE)
{ 
  if (any(sigma < 0))  stop(paste("sigma must be positive", "\n", "")) 
  if (any(nu <= 0))  stop(paste("nu must be positive", "\n", ""))
  if (any(tau < 0))  stop(paste("tau must be positive", "\n", ""))  
  if (log.p==TRUE) p <- exp(p) else p <- p
  if (any(p <= 0)|any(p >= 1))  stop(paste("p must be between 0 and 1", "\n", ""))       
  if (lower.tail==TRUE) p <- p else p <- 1-p
  suppressWarnings(q1 <- mu+(sigma/nu)*qt(p*(1+nu^2)/2, df=tau))
  suppressWarnings(q2 <- mu+(sigma*nu)*qt((p*(1+nu^2)-1)/(2*nu^2) + 0.5, df=tau))
  q <- ifelse(p < (1/(1+nu^2)), q1, q2)
  q
}
#-----------------------------------------------------------------  
rST3 <- function(n, mu=0, sigma=1, nu=1, tau=10)
{
  if (any(sigma <= 0))  stop(paste("sigma must be positive", "\n", "")) 
  if (any(nu <= 0))  stop(paste("nu must be positive", "\n", ""))
  if (any(tau <= 0))  stop(paste("tau must be positive", "\n", ""))  
  if (any(n <= 0))  stop(paste("n must be a positive integer", "\n", ""))    
  n <- ceiling(n)
  p <- runif(n)
  r <- qST3(p,mu=mu,sigma=sigma,nu=nu,tau=tau)
  r
}