## ----setup, echo=FALSE, message=FALSE, warning=FALSE--------------------------
require("fitdistrplus")
require("knitr") # for kable() function
set.seed(123) # here just to make random sampling reproducible
options(digits = 3)

## ----echo=TRUE, eval=FALSE----------------------------------------------------
# fitbench <- function(data, distr, method, grad = NULL,
#                      control = list(trace = 0, REPORT = 1, maxit = 1000),
#                      lower = -Inf, upper = +Inf, ...)

## ----echo=FALSE---------------------------------------------------------------
fitbench <- fitdistrplus:::fitbench

## -----------------------------------------------------------------------------
lnL <- function(par, fix.arg, obs, ddistnam) 
  fitdistrplus:::loglikelihood(par, fix.arg, obs, ddistnam) 
grlnlbeta <- fitdistrplus:::grlnlbeta

## ----fig.height=4, fig.width=4------------------------------------------------
#(1) beta distribution
n <- 200
x <- rbeta(n, 3, 3/4)
grlnlbeta(c(3, 4), x) #test
hist(x, prob=TRUE, xlim=0:1)
lines(density(x), col="red")
curve(dbeta(x, 3, 3/4), col="green", add=TRUE)
legend("topleft", lty=1, col=c("red","green"), legend=c("empirical", "theoretical"), bty="n")

## -----------------------------------------------------------------------------
ctr <- list(trace=0, REPORT=1, maxit=1000)

## -----------------------------------------------------------------------------
unconstropt <- fitbench(x, "beta", "mle", grad=grlnlbeta, lower=0)

## -----------------------------------------------------------------------------
dbeta2 <- function(x, shape1, shape2, log)
  dbeta(x, exp(shape1), exp(shape2), log=log)
#take the log of the starting values
startarg <- lapply(fitdistrplus:::startargdefault(x, "beta"), log)
#redefine the gradient for the new parametrization
grbetaexp <- function(par, obs, ...) 
    grlnlbeta(exp(par), obs) * exp(par)
    

expopt <- fitbench(x, distr="beta2", method="mle", grad=grbetaexp, start=startarg) 
#get back to original parametrization
expopt[c("fitted shape1", "fitted shape2"), ] <- exp(expopt[c("fitted shape1", "fitted shape2"), ])

## ----results='asis', echo=FALSE-----------------------------------------------
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3,
      caption="Unconstrained optimization with approximated gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3,
      caption="Unconstrained optimization with true gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3,
      caption="Exponential trick optimization with approximated gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(expopt[, grep("G-", colnames(expopt))], digits=3,
      caption="Exponential trick optimization with true gradient")

## ----fig.width=4, fig.height=4------------------------------------------------
llsurface(min.arg=c(0.1, 0.1), max.arg=c(7, 3), xlim=c(.1,7), 
          plot.arg=c("shape1", "shape2"), nlev=25,
          lseq=50, data=x, distr="beta", back.col = FALSE)
points(unconstropt[1,"BFGS"], unconstropt[2,"BFGS"], pch="+", col="red")
points(3, 3/4, pch="x", col="green")

## ----fig.width=4, fig.height=4------------------------------------------------
b1 <- bootdist(fitdist(x, "beta", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)
plot(b1, trueval = c(3, 3/4))

## -----------------------------------------------------------------------------
grlnlNB <- function(x, obs, ...)
{
  m <- x[1]
  p <- x[2]
  n <- length(obs)
  c(sum(psigamma(obs+m)) - n*psigamma(m) + n*log(p),
    m*n/p - sum(obs)/(1-p))
}

## ----fig.height=4, fig.width=4------------------------------------------------
#(2) negative binomial distribution
n <- 200
trueval <- c("size"=10, "prob"=3/4, "mu"=10/3)
x <- rnbinom(n, trueval["size"], trueval["prob"])

hist(x, prob=TRUE, ylim=c(0, .3), xlim=c(0, 10))
lines(density(x), col="red")
points(min(x):max(x), dnbinom(min(x):max(x), trueval["size"], trueval["prob"]), 
       col = "green")
legend("topright", lty = 1, col = c("red", "green"), 
       legend = c("empirical", "theoretical"), bty="n")

## -----------------------------------------------------------------------------
ctr <- list(trace = 0, REPORT = 1, maxit = 1000)
unconstropt <- fitbench(x, "nbinom", "mle", grad = grlnlNB, lower = 0)
unconstropt <- rbind(unconstropt, 
                     "fitted prob" = unconstropt["fitted mu", ] / (1 + unconstropt["fitted mu", ]))

## -----------------------------------------------------------------------------
dnbinom2 <- function(x, size, prob, log)
  dnbinom(x, exp(size), 1 / (1 + exp(-prob)), log = log)
# transform starting values
startarg <- fitdistrplus:::startargdefault(x, "nbinom")
startarg$mu <- startarg$size / (startarg$size + startarg$mu)
startarg <- list(size = log(startarg[[1]]), 
                 prob = log(startarg[[2]] / (1 - startarg[[2]])))

# redefine the gradient for the new parametrization
Trans <- function(x)
  c(exp(x[1]), plogis(x[2]))
grNBexp <- function(par, obs, ...) 
    grlnlNB(Trans(par), obs) * c(exp(par[1]), plogis(x[2])*(1-plogis(x[2])))

expopt <- fitbench(x, distr="nbinom2", method="mle", grad=grNBexp, start=startarg) 
# get back to original parametrization
expopt[c("fitted size", "fitted prob"), ] <- 
  apply(expopt[c("fitted size", "fitted prob"), ], 2, Trans)

## ----echo=FALSE---------------------------------------------------------------
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3,
      caption="Unconstrained optimization with approximated gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3,
      caption="Unconstrained optimization with true gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3,
      caption="Exponential trick optimization with approximated gradient")

## ----results='asis', echo=FALSE-----------------------------------------------
kable(expopt[, grep("G-", colnames(expopt))], digits=3,
      caption="Exponential trick optimization with true gradient")

## ----fig.width=4, fig.height=4------------------------------------------------
llsurface(min.arg = c(5, 0.3), max.arg = c(15, 1), xlim=c(5, 15),
          plot.arg = c("size", "prob"), nlev = 25,
          lseq = 50, data = x, distr = "nbinom", back.col = FALSE)
points(unconstropt["fitted size", "BFGS"], unconstropt["fitted prob", "BFGS"], 
       pch = "+", col = "red")
points(trueval["size"], trueval["prob"], pch = "x", col = "green")

## ----fig.width=4, fig.height=4------------------------------------------------
b1 <- bootdist(fitdist(x, "nbinom", method = "mle", optim.method = "BFGS"), 
               niter = 100, parallel = "snow", ncpus = 2)
summary(b1)
plot(b1, trueval=trueval[c("size", "mu")]) 

