Fitted distributions

knitr::opts_chunk$set(echo=FALSE, warning=FALSE, message=FALSE,
                      fig.pos = 'h',
                      fig.align = 'center',
                      fig.height = 3,
                      fig.width = 4)

Elicited cumulative probabilities

mydf <- data.frame( fit$vals[expert, ], fit$probs[expert, ])
colnames(mydf) <- c( "$x$", "$P(X\\le x)$")
knitr::kable(mydf)
if(fit$limits[expert, 1] == 0){
  x <- paste0("x")
  lower <- "`"}else{
    lower <- paste0("`", fit$limits[expert, 1], " + ")}

if(fit$limits[expert, 1] > 0){
  x <- paste0("(x-", fit$limits[expert, 1],")")}
if(fit$limits[expert, 1] < 0){
  x <- paste0("(x+", abs(fit$limits[expert, 1]),")")}

if(fit$limits[expert, 2] < Inf){
  xMirror <- paste0("(", fit$limits[expert, 2],"-x)")
  upper <- paste0("`", fit$limits[expert, 2], " - ")
}
lx <- min(fit$vals[expert, ])
ux <- max(fit$vals[expert, ])

Distributions

All parameter values reported to r sf significant figures.

Normal

plotfit(fit, ex = expert, d = "normal")
mu <- signif(fit$Normal[expert, 1], sf)
sigsq <- signif(fit$Normal[expert, 2]^2, sf)

$$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2 \sigma^2}(x - \mu)^2\right),\quad -\infty<x<\infty, $$ with \begin{align} \mu &= r mu,\ \sigma^2 &= r sigsq. \end{align}

Sample n = 1000 random values with the command

rnorm(n = 1000, mean =r mu, sd = sqrt(r sigsq))


Student$-t$

m <- signif(fit$Student.t[expert, 1], 3)
s <- signif(fit$Student.t[expert, 2], 3)
tdf <- fit$Student.t[expert, 3]
plotfit(fit, ex = expert, d = "t")

$$ f_X(x) = \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\sigma\sqrt{\nu \pi}} \left(1 + \frac{1}{\nu}\left(\frac{x - \mu}{\sigma}\right)^2\right)^{-(\nu + 1)/2},\quad -\infty<x<\infty, $$ with \begin{align} \nu &= r tdf,\ \mu &= r m,\ \sigma &= r s.\ \end{align}

Sample n = 1000 random values with the command

r m+r s* rt(n = 1000, df =r tdf)


Log normal

if(is.na(fit$ssq[expert, 4])){
  equationLogNormal <- "Distribution not fitted. Argument `lower` needs to be specified in the `fitdist` command."
  samplingLogNormal <- NULL
  samplingLogNormalText <- NULL
  shiftLognormal <- NULL
}else{
  mu <- signif(fit$Log.normal[expert, 1], 3)
  sigsq <- signif(fit$Log.normal[expert, 2]^2, 3)  

  equationLogNormal <- paste0("$$f_X(x) = \\frac{1}{", x, "} \\times \\frac{1}{\\sqrt{2\\pi\\sigma^2}} 
\\exp\\left(-\\frac{1}{2\\sigma^2}(\\ln ", x,"  - \\mu)^2\\right), \\quad x >", fit$limits[expert, 1],"$$ and $f_X(x)=0$ otherwise, 
with $$\\mu = ", mu,"$$ $$\\sigma^2  = ", sigsq, "$$")



  samplingLogNormal <- paste0(lower, " rlnorm(n = 1000, meanlog = ", mu,", sdlog = sqrt(",
sigsq,"))`" )

  samplingLogNormalText <- "Sample `n = 1000` random values with the command"

  if(fit$limits[expert, 1] != 0){
    shiftLognormal <- paste0("Log normal distribution shifted to have support over the interval $(",
                               fit$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLognormal <- NULL
  }

} 

r paste(shiftLognormal)

if(!is.na(fit$ssq[expert, 4])){
plotfit(fit, ex = expert, d = "lognormal")
}

r paste(equationLogNormal)

r paste(samplingLogNormalText)

r paste(samplingLogNormal)


Gamma

if(is.na(fit$ssq[expert, 3])){
  equationGamma <- "Distribution not fitted. Argument `lower` needs to be specified in the `fitdist` command."
  samplingGamma <- NULL
  samplingGammaText <- NULL
  shiftGamma <- NULL
}else{
  shape <- signif(fit$Gamma[expert, 1], 3)
  rate <- signif(fit$Gamma[expert, 2], 3)  

  equationGamma <- paste0("$$f_X(x) =\\frac{\\beta ^ {\\alpha}}{\\Gamma(\\alpha)}",  x,
"^{\\alpha - 1} \\exp\\left(- \\beta ", x,"\\right), \\quad x >", fit$limits[expert, 1],"$$ and $f_X(x)=0$ otherwise, 
with $$\\alpha = ", shape,"$$ $$\\beta  = ", rate, "$$")



  samplingGamma <- paste0(lower, " rgamma(n = 1000, shape = ", shape,", rate = ",
rate,")`" )

  samplingGammaText <- "Sample `n = 1000` random values with the command"

   if(fit$limits[expert, 1] != 0){
    shiftGamma <- paste0("Gamma distribution shifted to have support over the interval $(",
                               fit$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftGamma <- NULL
  }

} 

r paste(shiftGamma)

if(!is.na(fit$ssq[expert, 3])){
plotfit(fit, ex = expert, d = "gamma")
}

r paste(equationGamma)

r paste(samplingGammaText)

r paste(samplingGamma)


Log Student$-t$

if(is.na(fit$ssq[expert, 4])){
  equationLogStudentT <- "Distribution not fitted. Argument `lower` needs to be specified in the `fitdist` command."
  samplingLogStudentT <- NULL
  samplingLogStudentTText <- NULL
  shiftLogStudentT <- NULL
}else{
  m <- signif(fit$Log.Student.t[expert, 1], 3)
  s <- signif(fit$Log.Student.t[expert, 2], 3)
  tdf <- fit$Log.Student.t[expert, 3]

  equationLogStudentT <- paste0("$$f_X(x) =\\frac{1}{", x, "} \\times \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\times\\sigma\\times \\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{\\ln ", x, " - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2}, \\quad x >", fit$limits[expert, 1],"$$ and $f_X(x)=0$ otherwise, 
with $$\\nu = ", tdf,"$$ $$\\mu  = ", m, "$$ $$\\sigma = ", s , ".$$")



  samplingLogStudentT <- paste0(lower, " exp(", m, " + ", s, " * rt(n = 1000, df = ", tdf,"))`" )

  samplingLogStudentTText <- "Sample `n = 1000` random values with the command"

  if(fit$limits[expert, 1] != 0){
    shiftLogStudentT <- paste0("Log Student-$t$ distribution shifted to have support over the interval $(",
                               fit$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLogStudentT <- NULL
  }
} 

r paste(shiftLogStudentT)

if(!is.na(fit$ssq[expert, "logt"])){
plotfit(fit, ex = expert, d = "logt")
}

r paste(equationLogStudentT)

r paste(samplingLogStudentTText)

r paste(samplingLogStudentT)


Beta

if(fit$limits[expert, 1] == 0)
  x <- paste0("x")
if(fit$limits[expert, 1] > 0)
  x <- paste0("x-", fit$limits[1, expert])
if(fit$limits[expert, 1] < 0)
  x <- paste0("x+", abs(fit$limits[1, expert]))
if(fit$limits[expert, 2] < Inf & fit$limits[expert, 1] > -Inf){
  r <- fit$limits[expert, 2] - fit$limits[expert, 1]
  if(r !=1){
  mult = paste0(r, " *")}else{
    mult = ""
  }
}


shape1 <- signif(fit$Beta[expert, 1], 3)
shape2 <- signif(fit$Beta[expert, 2], 3)
if(is.na(fit$ssq[expert, 6])){
  scalingBeta <- NULL
  equationBeta <- "Distribution not fitted. Arguments `lower` and `upper` need to be specified in the `fitdist` command."
  samplingBeta <- NULL
  samplingBetaText <- NULL
}else{
  if(r == 1 & fit$limits[expert, 1] == 0){
  equationBeta <- paste0("$$f_X(x) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}x ^{\\alpha - 1} \\left(1 - x\\right)^{\\beta - 1}, \\quad  0 < x < 1, $$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
  scalingBeta <- NULL
  }
  if(r == 1 & fit$limits[expert, 1] != 0){
    equationBeta <- paste0("$$f_X(x) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(", x,"\\right) ^{\\alpha - 1} \\left(1 - \\left(", x,"\\right)\\right)^{\\beta - 1}, \\quad  ", fit$limits[expert, 1]," < x <", fit$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          fit$limits[expert, 1],
                          ", ",
                          fit$limits[expert, 2],
                          "].")
  }


  if(r !=1){
    equationBeta <- paste0("$$f_X(x) = \\frac{1}{", r, "}\\times\\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(\\frac{", x,"}{", r,"}\\right) ^{\\alpha - 1} \\left(1 - \\left(\\frac{", x,"}{", r,"}\\right)\\right)^{\\beta - 1}, \\quad  ", fit$limits[expert, 1]," < x <", fit$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          fit$limits[expert, 1],
                          ", ",
                          fit$limits[expert, 2],
                          "].")

  }

  samplingBeta <- paste0(lower, mult, " rbeta(n = 1000, shape1 = ", shape1, ", shape2 = ", shape2, ")`")

  samplingBetaText <- "Sample `n = 1000` random values with the command"
} 

r paste(scalingBeta)

if(!is.na(fit$ssq[expert, "beta"])){
plotfit(fit, ex = expert, d = "beta")
}

r paste(equationBeta)

r paste(samplingBetaText)

r paste(samplingBeta)

Mirror gamma

if(is.na(fit$ssq[expert, "mirrorgamma"])){
  equationMirrorGamma <- "Distribution not fitted. Argument `upper` needs to be specified in the `fitdist` command."
  samplingMirrorGamma <- NULL
  samplingMirrorGammaText <- NULL
  shiftMirrorGamma <- NULL
}else{
  shape <- signif(fit$mirrorgamma[expert, 1], 3)
  rate <- signif(fit$mirrorgamma[expert, 2], 3)  

  equationMirrorGamma <- paste0("$$f_X(x) =\\frac{\\beta ^ {\\alpha}}{\\Gamma(\\alpha)}",  xMirror,
"^{\\alpha - 1} \\exp\\left(- \\beta ", xMirror,"\\right), \\quad x <", fit$limits[expert, 2],"$$ and $f_X(x)=0$ otherwise, 
with $$\\alpha = ", shape,"$$ $$\\beta  = ", rate, "$$")



  samplingMirrorGamma <- paste0(upper, " rgamma(n = 1000, shape = ", shape,", rate = ",
rate,")`" )

  samplingMirrorGammaText <- "Sample `n = 1000` random values with the command"


    shiftMirrorGamma <- paste0("Gamma distribution fitted to ", fit$limits[expert, 2]," - $X$.")


} 

r paste(shiftMirrorGamma)

if(!is.na(fit$ssq[expert, "mirrorgamma"])){
plotfit(fit, ex = expert, d = "mirrorgamma")
}

r paste(equationMirrorGamma)

r paste(samplingMirrorGammaText)

r paste(samplingMirrorGamma)


Mirror Log normal

if(is.na(fit$ssq[expert, "mirrorlognormal"])){
  equationMirrorLogNormal <- "Distribution not fitted. Argument `upper` needs to be specified in the `fitdist` command."
  samplingMirrorLogNormal <- NULL
  samplingMirrorLogNormalText <- NULL
  shiftMirrorLognormal <- NULL
}else{
  mu <- signif(fit$mirrorlognormal[expert, 1], 3)
  sigsq <- signif(fit$mirrorlognormal[expert, 2]^2, 3)  

  equationMirrorLogNormal <- paste0("$$f_X(x) = \\frac{1}{", xMirror, "} \\times \\frac{1}{\\sqrt{2\\pi\\sigma^2}} 
\\exp\\left(-\\frac{1}{2\\sigma^2}(\\ln ", xMirror,"  - \\mu)^2\\right), \\quad x <", fit$limits[expert, 2],"$$ and $f_X(x)=0$ otherwise, 
with $$\\mu = ", mu,"$$ $$\\sigma^2  = ", sigsq, "$$")



  samplingMirrorLogNormal <- paste0(upper, " rlnorm(n = 1000, meanlog = ", mu,", sdlog = sqrt(",
sigsq,"))`" )

  samplingMirrorLogNormalText <- "Sample `n = 1000` random values with the command"

   shiftMirrorLognormal <- paste0("Normal distribution fitted to $\\log$ (  ", fit$limits[expert, 2]," - $X$).")

} 

r paste(shiftMirrorLognormal)

if(!is.na(fit$ssq[expert, "mirrorlognormal"])){
plotfit(fit, ex = expert, d = "mirrorlognormal")
}

r paste(equationMirrorLogNormal)

r paste(samplingMirrorLogNormalText)

r paste(samplingMirrorLogNormal)


Mirror log Student$-t$

if(is.na(fit$ssq[expert, "mirrorlogt"])){
  equationMirrorLogStudentT <- "Distribution not fitted. Argument `lower` needs to be specified in the `fitdist` command."
  samplingMirrorLogStudentT <- NULL
  samplingMirrorLogStudentTText <- NULL
  shiftMirrorLogStudentT <- NULL
}else{
  m <- signif(fit$mirrorlogt[expert, 1], 3)
  s <- signif(fit$mirrorlogt[expert, 2], 3)
  tdf <- fit$mirrorlogt[expert, 3]

  equationMirrorLogStudentT <- paste0("$$f_X(x) =\\frac{1}{", xMirror, "} \\times \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\times\\sigma\\times \\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{\\ln ", xMirror, " - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2}, \\quad x <", fit$limits[expert, 2],"$$ and $f_X(x)=0$ otherwise, 
with $$\\nu = ", tdf,"$$ $$\\mu  = ", m, "$$ $$\\sigma = ", s , ".$$")



  samplingMirrorLogStudentT <- paste0(upper, " exp(", m, " + ", s, " * rt(n = 1000, df = ", tdf,"))`" )

  samplingMirrorLogStudentTText <- "Sample `n = 1000` random values with the command"

   shiftMirrorLogStudentT <- paste0("Student-$t$ distribution fitted to $\\log$ (  ", fit$limits[expert, 2]," - $X$).")

} 

r paste(shiftMirrorLogStudentT)

if(!is.na(fit$ssq[expert, "mirrorlogt"])){
plotfit(fit, ex = expert, d = "mirrorlogt")
}

r paste(equationMirrorLogStudentT)

r paste(samplingMirrorLogStudentTText)

r paste(samplingMirrorLogStudentT)



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SHELF documentation built on June 7, 2023, 5:11 p.m.