R/ZASICHEL.R

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

# Created bt Mikis Stasinopoulos and Bob Rigby 
# May 2017
# Zero adjusted Sichel
#----------------------------------------------------------------------------------------
ZASICHEL <-function (mu.link ="log", sigma.link="log", nu.link="identity", tau.link = "logit") 
{
    mstats <- checklink("mu.link", "zaSICHEL", substitute(mu.link), 
                         c("1/mu^2", "log", "identity"))
    dstats <- checklink("sigma.link", "zaSICHEL", substitute(sigma.link), 
                         c("inverse", "log", "identity"))
    vstats <- checklink("nu.link", "zaSICHEL",substitute(nu.link), 
                         c("1/nu^2", "log", "identity"))  
    tstats <- checklink("tau.link", "zaSICHEL", substitute(tau.link), 
                        c("logit", "probit", "cloglog", "log", "own"))
    structure(
          list(family = c("ZASICHEL", "zero adjusted Sichel"),
           parameters = list(mu = TRUE, sigma = TRUE, nu = TRUE, tau=TRUE), 
                nopar = 4, 
                 type = "Discrete",
              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) 
        {
          dldm0 <- SICHEL()$dldm(y,mu,sigma,nu) + dSICHEL(0, mu,sigma,nu)*SICHEL()$dldm(0,mu,sigma,nu)/(1-dSICHEL(0,mu,sigma,nu))
          dldm <- ifelse(y==0, 0 , dldm0)
          dldm }, 
        d2ldm2 = function(y,mu,sigma,nu,tau) 
        {
          #dldm0 <-    NBI()$dldm(y,mu,sigma)    +    dNBI(0 ,mu,sigma)*      NBI()$dldm(0,mu,sigma   )/(1-   dNBI(0,mu,sigma   ))
          dldm0 <- SICHEL()$dldm(y,mu,sigma,nu) + dSICHEL(0, mu,sigma,nu)*SICHEL()$dldm(0,mu,sigma,nu)/(1-dSICHEL(0,mu,sigma,nu))
          dldm <- ifelse(y==0, 0 , dldm0)
          d2ldm2 <- -dldm*dldm
          d2ldm2 <- ifelse(d2ldm2 < -1e-15, d2ldm2,-1e-15)    
          d2ldm2},
        dldd = function(y,mu,sigma,nu,tau) 
        { 
          sigma <- ifelse(sigma<0.000001, 0.000001, sigma )
          dldd0 <- SICHEL()$dldd(y,mu,sigma, nu) + dSICHEL(0,mu,sigma,nu)*SICHEL()$dldd(0,mu,sigma, nu)/(1-dSICHEL(0,mu,sigma, nu))
          dldd <- ifelse(y==0, 0 , dldd0)
          dldd },
        d2ldd2 = function(y,mu,sigma,nu, tau)
        {
          sigma <- ifelse(sigma<0.000001, 0.000001, sigma )
          dldd0 <- SICHEL()$dldd(y,mu,sigma, nu) + dSICHEL(0,mu,sigma,nu)*SICHEL()$dldd(0,mu,sigma, nu)/(1-dSICHEL(0,mu,sigma, nu))
          dldd <- ifelse(y==0, 0 , dldd0)
          d2ldd2 <- -dldd*dldd
          d2ldd2 <- ifelse(d2ldd2 < -1e-15, d2ldd2,-1e-15)  
          d2ldd2 },
        dldv = function(y,mu,sigma,nu, tau) 
        {
          nd <- numeric.deriv(dZASICHEL(y, mu, sigma, nu, tau, log=TRUE), "nu", delta=0.001)
          dldv <- as.vector(attr(nd, "gradient"))
          dldv },
        
        d2ldv2 = function(y,mu,sigma,nu, tau){
          nd <- numeric.deriv(dZASICHEL(y, mu, sigma, nu, tau, log=TRUE), "nu", delta=0.001)
          dldv <- as.vector(attr(nd, "gradient"))
          d2ldv2 <- -dldv*dldv
          d2ldv2 <- ifelse(d2ldv2 < -1e-15, d2ldv2,-1e-15)  
          d2ldv2 },
        dldt = function(y,mu,sigma,nu, tau) 
        {
          #cat("at tau ", mu[1], sigma[1], nu[1], tau[1], "\n")
          dldt <- ifelse(y==0, 1/tau, -1/(1-tau))
          dldt }, 
        d2ldt2 = function(y,mu,sigma,nu, tau){
          d2ldt2 <- -1/(tau*(1-tau))# or -1/((1-tau)^2)
          d2ldt2 <- ifelse(d2ldt2 < -1e-15, d2ldt2,-1e-15)  
          d2ldt2},
        d2ldmdd = function(y,mu,sigma,nu, tau) #1
        {
          dldm0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldm(0,mu,sigma,nu)
          dldm <- ifelse(y==0, dldm0, SICHEL()$dldm(y,mu,sigma,nu))             
          dldd0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldd(0,mu,sigma,nu)
          dldd <- ifelse(y==0, dldd0, SICHEL()$dldd(y,mu,sigma,nu))
          d2ldmdd <- -dldm *dldd
          d2ldmdd <- ifelse( d2ldmdd < -1e-15,  d2ldmdd,-1e-15)  
          d2ldmdd }, 
        d2ldmdv = function(y, mu, sigma, nu, tau) # 2
        {
          dldm0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldm(0,mu,sigma,nu)
          dldm <- ifelse(y==0, dldm0, SICHEL()$dldm(y,mu,sigma,nu))          
          nd <- numeric.deriv(dSICHEL(y, mu, sigma, nu, log=TRUE), "nu", delta=0.001)
          dldv <- as.vector(attr(nd, "gradient"))
          d2ldmdv <- -dldm *dldv
          d2ldmdv <- ifelse(d2ldmdv < -1e-15, d2ldmdv,-1e-15)  
          d2ldmdv }, 
        d2ldmdt = function(y, mu, sigma, nu, tau) # 3
        {
          dldm0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldm(0,mu,sigma,nu)
          dldm <- ifelse(y==0, dldm0, SICHEL()$dldm(y,mu,sigma,nu))          
          dldt0 <- ((tau+(1-tau)*dSICHEL(0,mu,sigma, nu))^(-1))*(1-dSICHEL(0,mu,sigma, nu))
          dldt <- ifelse(y==0, dldt0, -1/(1-tau))
          d2ldmdt <- -dldm *dldt
          d2ldmdt <- ifelse(d2ldmdt < -1e-15, d2ldmdt,-1e-15)  
          d2ldmdt }, 
        d2ldddv = function(y,mu,sigma,nu,tau) # 4
        {
          dldd0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldd(0,mu,sigma,nu)
          dldd <- ifelse(y==0, dldd0, SICHEL()$dldd(y,mu,sigma,nu))
          nd <- numeric.deriv(dZASICHEL(y, mu, sigma, nu, tau, log=TRUE), "nu", delta=0.001)
          dldv <- as.vector(attr(nd, "gradient"))
          d2ldddv <- -dldd *dldv
          d2ldddv <- ifelse(d2ldddv < -1e-15, d2ldddv,-1e-15) 
          d2ldddv },               
        d2ldddt = function(y,mu,sigma,nu,tau) # 5
        {
          dldd0 <- (1-tau)*((tau+(1-tau)*dSICHEL(0,mu,sigma,nu))^(-1))*dSICHEL(0,mu,sigma,nu)*SICHEL()$dldd(0,mu,sigma,nu)
          dldd <- ifelse(y==0, dldd0, SICHEL()$dldd(y,mu,sigma,nu))
          dldt0 <- ((tau+(1-tau)*dSICHEL(0,mu,sigma, nu))^(-1))*(1-dSICHEL(0,mu,sigma, nu))
          dldt <- ifelse(y==0, dldt0, -1/(1-tau))
          d2ldddt <- -dldd *dldt
          d2ldddt <- ifelse(d2ldddt < -1e-15, d2ldddt,-1e-15) 
          d2ldddt },   
        d2ldvdt = function(y,mu,sigma,nu,tau) # 6
        {
          nd <- numeric.deriv(dSICHEL(y, mu, sigma, nu, log=TRUE), "nu", delta=0.001)
          dldv <- as.vector(attr(nd, "gradient"))
          dldt0 <- ((tau+(1-tau)*dSICHEL(0,mu,sigma, nu))^(-1))*(1-dSICHEL(0,mu,sigma, nu))
          dldt <- ifelse(y==0, dldt0, -1/(1-tau))
          d2ldvdt <- -dldv *dldt
          d2ldvdt <- ifelse(d2ldvdt < -1e-15, d2ldvdt,-1e-15) 
          d2ldvdt },      
        G.dev.incr  = function(y,mu,sigma,nu, tau,...) -2*dZASICHEL(y, mu, sigma, nu, tau, log=TRUE),
        rqres = expression(    
          rqres(pfun="pZASICHEL", type="Discrete", ymin=0, y=y, mu=mu, sigma=sigma, nu=nu)
        ), 
        mu.initial = expression(mu<- (y+mean(y))/2 ),
        sigma.initial = expression(
          sigma <- rep( max( ((var(y)-mean(y))/(mean(y)^2)),0.1),length(y))),
        nu.initial = expression({  nu <- rep(-0.5,length(y)) }), 
        tau.initial = expression({  tau <- rep(sum(y==0)/length(y),length(y))}), 
        mu.valid = function(mu) all(mu > 0) , 
        sigma.valid = function(sigma)  all(sigma > 0), 
        nu.valid = function(nu) TRUE, 
        tau.valid = function(nu) all(nu > 0 & nu < 1),  
        y.valid = function(y)  all(y >= 0),
        mean = function(mu, sigma, nu, tau)
        {
          p0 <- dSICHEL(0, mu, sigma, nu) 
          c  <- (1 - tau) / (1 - p0) 
          return( mu * c)
        },
        variance = function(mu, sigma, nu, tau) 
        {
          K1  <- besselK(1 / sigma, nu +1)
          K2  <- besselK(1 / sigma, nu)
          b   <- K1 / K2
          p0  <- dSICHEL(0, mu, sigma, nu) 
          c   <- (1 - tau) / (1 - p0)
          h1  <- 1 / c * (2 * sigma * (nu + 1) / b + 1 / b^2) - 1
          return(c * mu + c^2 * mu^2 * h1)
        }
          ),
        class = c("gamlss.family","family"))
}
#----------------------------------------------------------------------------------------
#------------------------------------------------------------------------------------------
#------------------------------------------------------------------------------------------
dZASICHEL<-function(x, mu=1, sigma=1, nu=-0.5, tau=0.1, log=FALSE)
{
  if (any(mu <= 0) )  stop(paste("mu must be greater than 0 ", "\n", "")) 
  if (any(sigma <= 0) )  stop(paste("sigma must be greater than 0 ", "\n", "")) 
  if (any(tau <= 0)|any(tau >= 1))  stop(paste("tau must be between 0 and 1 ", "\n", ""))
 # if (any(x < 0) )  stop(paste("x must be >=0", "\n", "")) 
        ly <- max(length(x),length(mu),length(sigma),length(nu),length(tau)) 
         x <- rep(x, length = ly)      
    sigma <- rep(sigma, length = ly)
       mu <- rep(mu, length = ly)   
       nu <- rep(nu, length = ly) 
      tau <- rep(tau, length = ly)
      fy0 <- dSICHEL(0, mu = mu, sigma=sigma,  nu=nu)
       fy <- dSICHEL(x, mu = mu, sigma=sigma, nu=nu, log = TRUE)                   
    logfy <- rep(0, length(x))
    logfy <- ifelse((x==0), log(tau), log(1-tau) + fy - log(1-fy0))          
    if(log == FALSE) fy2 <- exp(logfy) else fy2 <- logfy
     fy2 <- ifelse(x < 0, 0, fy2) 
     fy2
  }
#----------------------------------------------------------------------------------------     
#-----------------------------------------------
pZASICHEL <- function(q, mu=1, sigma=1, nu=-0.5, tau=0.1, lower.tail = TRUE, log.p = FALSE)
{ 
   if (any(mu <= 0) )  stop(paste("mu must be greater than 0 ", "\n", "")) 
   if (any(sigma <= 0) )  stop(paste("sigma must be greater than 0 ", "\n", "")) 
   if (any(tau <= 0)|any(tau >= 1))  #In this parametrization  nu = alpha
      stop(paste("tau must be between 0 and 1 ", "\n", ""))
#   if (any(q < 0) )  stop(paste("y must be >=0", "\n", ""))
     ly <- max(length(q),length(mu),length(sigma),length(nu),length(tau)) 
      q <- rep(q, length = ly)      
  sigma <- rep(sigma, length = ly)
     mu <- rep(mu, length = ly)   
     nu <- rep(nu, length = ly) 
    tau <- rep(tau, length = ly)
   cdf0 <- pSICHEL(0, mu = mu, sigma=sigma, nu=nu)
   cdf1 <- pSICHEL(q, mu = mu, sigma=sigma, nu=nu)                   
   cdf3 <- tau+((1-tau)*(cdf1-cdf0)/(1-cdf0))
    cdf <- ifelse((q==0),tau,  cdf3)
  if(lower.tail == TRUE) cdf <- cdf else cdf <-1-cdf
  if(log.p==FALSE) cdf <- cdf else cdf <- log(cdf)   
  cdf <- ifelse(q < 0, 0, cdf) 
  cdf
}
#----------------------------------------------------------------------------------------
qZASICHEL <- function(p, mu=1, sigma=1, nu=-0.5, tau=0.1, lower.tail = TRUE, 
                      log.p = FALSE,  max.value=10000)
  {     
       if (any(mu <= 0) )  stop(paste("mu must be greater than 0 ", "\n", "")) 
       if (any(sigma <= 0) )  stop(paste("sigma must be greater than 0 ", "\n", ""))
       if (any(tau <= 0)|any(tau >= 1))  #In this parametrization  nu = alpha
              stop(paste("tau must be between 0 and 1 ", "\n", ""))
       if (any(p < 0) | any(p > 1))  stop(paste("p must be between 0 and 1", "\n", ""))    
       if (log.p == TRUE) p <- exp(p)   else p <- p
       if (lower.tail == TRUE)  p <- p  else p <- 1 - p
       ly <- max(length(p),length(mu),length(sigma),length(nu),length(tau)) 
            p <- rep(p, length = ly)      
        sigma <- rep(sigma, length = ly)
           mu <- rep(mu, length = ly)   
           nu <- rep(nu, length = ly) 
          tau <- rep(tau, length = ly)
         pnew <- (p-tau)/(1-tau)-1e-10
         cdf0 <- pSICHEL(0, mu = mu, sigma=sigma, nu=nu)                   
         pnew2 <- cdf0*(1-pnew) + pnew           
         pnew2 <- ifelse((pnew2 > 0 ),pnew2, 0)
             q <- qSICHEL(pnew2, mu = mu, sigma=sigma, nu=nu,  max.value= max.value)                   
             q
   }
#----------------------------------------------------------------------------------------
rZASICHEL <- function(n, mu=1, sigma=1, nu=-0.5, tau=0.1,  max.value=10000)
  { 
         if (any(mu <= 0) )  stop(paste("mu must be greater than 0 ", "\n", "")) 
         if (any(sigma <= 0) )  stop(paste("sigma must be greater than 0 ", "\n", "")) 
         if (any(tau <= 0)|any(tau >= 1))  #In this parametrization  nu = alpha
            stop(paste("tau must be between 0 and 1 ", "\n", ""))
         if (any(n <= 0))  stop(paste("n must be a positive integer", "\n", ""))  
          n <- ceiling(n)
          p <- runif(n)
          r <- qZASICHEL(p, mu=mu, sigma=sigma, nu=nu, tau=tau,  max.value= max.value)
          as.integer(r)
  }