In OakleyJ/SHELF: Tools to Support the Sheffield Elicitation Framework
if(roulette == TRUE){
bins <- paste0("(",bin.left,", ",bin.right,"]")
cat("Distribution elicited using the roulette method.")
knitr::kable(data.frame(bins = bins, probs = chips))
}
Elicited cumulative probabilities
```{asis, echo = roulette}
Implied cumulative probabilities used for distribution fitting are as follows
```r
expert <- 1
sf <- 3
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]),")")}
xMirror <- paste0("(", fit$limits[expert, 2],"-x)")
upper <- paste0("`", fit$limits[expert, 2], " - ")
lx <- min(fit$vals[expert, ])
ux <- max(fit$vals[expert, ])
judgementsDF <- data.frame(judgement = c("lower limit",
"upper limit",
"smallest elicited probability",
"largest elicited probability"
)
,
value = c(fit$limits[expert, "lower"],
fit$limits[expert, "upper"],
min(fit$probs[expert, ]),
max(fit$probs[expert, ])
)
)
```{asis, echo = any(is.na(fit$ssq[expert, ]))}
Note: not all distributions have been fitted. The requirements for fitting each distribution are as follows.
-
Normal, Student-t, skew normal: smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Log normal and log Student-t: finite lower limit; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Gamma: finite lower limit; at least one elicited probability between 0 and 1.
-
Beta : finite lower and upper limits; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Mirror gamma: finite upper limit; at least one elicited probability between 0 and 1.
-
Mirror log normal and mirror log Student-t: finite upper limit; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
The judgements provided were as follows.
```r
knitr::kable(judgementsDF)
Distributions
All parameter values reported to 3 significant figures.
Normal
if(is.na(fit$ssq[expert, "normal"])){
equationNormal <- "Distribution not fitted."
samplingNormal <- NULL
samplingNormalText <- NULL
}else{
mu <- signif(fit$Normal[expert, 1], sf)
sigsq <- signif(fit$Normal[expert, 2]^2, sf)
equationNormal <- paste0("$$f_X(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}
\\exp\\left(-\\frac{1}{2 \\sigma^2}(x - \\mu)^2\\right),\\quad -\\infty<x<\\infty,$$
with $$\\mu = ", mu,"$$ $$\\sigma^2 = ", sigsq, "$$")
samplingNormal <- paste0("` rnorm(n = 1000, mean = ", mu,", sd = sqrt(",
sigsq,"))`" )
samplingNormalText <- "Sample `n = 1000` random values with the command"
}
if(!is.na(fit$ssq[expert, "normal"])){
plotfit(fit, d = "normal")
}
r paste(equationNormal)
r paste(samplingNormalText)
r paste(samplingNormal)
Student$-t$
if(is.na(fit$ssq[expert, "t"])){
equationT <- "Distribution not fitted."
samplingT <- NULL
samplingTtext <- NULL
}else{
m <- signif(fit$Student.t[expert, 1], 3)
s <- signif(fit$Student.t[expert, 2], 3)
tdf <- fit$Student.t[expert, 3]
equationT <- paste0("$$
f_X(x) = \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\sigma\\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{x - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2},\\quad -\\infty<x<\\infty,
$$
with $$\\nu = ", tdf,"$$ $$\\mu = ", m,"$$ $$\\sigma = ", s, "$$")
samplingT <- paste0("`", m, " + ", s, " * rt(n = 1000, df = ", tdf, ")`")
samplingTtext <- "Sample `n = 1000` random values with the command"
}
r paste(equationT)
r paste(samplingTtext)
r paste(samplingT)
Skew normal
if(is.na(fit$ssq[expert, "skewnormal"])){
equationSkewNormal <- "Distribution not fitted."
samplingSkewNormal <- NULL
samplingSkewNormalText <- NULL
}else{
xi <- signif(fit$Skewnormal[expert, 1], sf)
omega <- signif(fit$Skewnormal[expert, 2], sf)
alpha <- signif(fit$Skewnormal[expert, 3], sf)
equationSkewNormal <- paste0("$$f_X(x)=\\frac{2}{\\omega}\\phi\\left(\\frac{x-\\xi}{\\omega}\\right)\\Phi\\left(\\alpha\\left(\\frac{x-\\xi}{\\omega}\\right)\\right),\\quad -\\infty<x<\\infty,$$
where $\\phi(.)$ and $\\Phi(.)$ are the probability density function and cumulative distribution function respectively of the standard normal distribution, and with $$\\xi = ", xi,"$$ $$\\omega = ", omega, "$$ $$\\alpha = ", alpha, "$$")
samplingSkewNormal <- paste0("` sn::rsn(n = 1000, xi = ", xi,", omega = ",
omega, ", alpha = ",alpha,"))`" )
samplingSkewNormalText <- "Sample `n = 1000` random values with the command"
}
if(!is.na(fit$ssq[expert, "skewnormal"])){
plotfit(fit, d = "skewnormal")
}
r paste(equationSkewNormal)
r paste(samplingSkewNormalText)
r paste(samplingSkewNormal)
Log normal
if(is.na(fit$ssq[expert, "lognormal"])){
equationLogNormal <- "Distribution not fitted."
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, "lognormal"])){
plotfit(fit, d = "lognormal")
}
r paste(equationLogNormal)
r paste(samplingLogNormalText)
r paste(samplingLogNormal)
Gamma
if(is.na(fit$ssq[expert, "gamma"])){
equationGamma <- "Distribution not fitted."
samplingGamma <- NULL
samplingGammaText <- NULL
shiftGamma <- NULL
}else{
shape <- signif(fit$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, "gamma"])){
plotfit(fit, d = "gamma")
}
r paste(equationGamma)
r paste(samplingGammaText)
r paste(samplingGamma)
Log Student$-t$
if(is.na(fit$ssq[expert, "logt"])){
equationLogStudentT <- "Distribution not fitted."
samplingLogStudentT <- NULL
samplingLogStudentTText <- NULL
shiftLogStudentT <- NULL
}else{
m <- signif(fit$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, 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, "beta"])){
scalingBeta <- NULL
equationBeta <- "Distribution not fitted."
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$ssq[expert, "beta"])){
plotfit(fit, d = "beta")
}
r paste(equationBeta)
r paste(samplingBetaText)
r paste(samplingBeta)
Mirror gamma
if(is.na(fit$ssq[expert, "mirrorgamma"])){
equationMirrorGamma <- "Distribution not fitted."
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$ssq[expert, "mirrorgamma"])){
plotfit(fit, d = "mirrorgamma")
}
r paste(equationMirrorGamma)
r paste(samplingMirrorGammaText)
r paste(samplingMirrorGamma)
Mirror Log normal
if(is.na(fit$ssq[expert, "mirrorlognormal"])){
equationMirrorLogNormal <- "Distribution not fitted."
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$ssq[expert, "mirrorlognormal"])){
plotfit(fit, d = "mirrorlognormal")
}
r paste(equationMirrorLogNormal)
r paste(samplingMirrorLogNormalText)
r paste(samplingMirrorLogNormal)
Mirror log Student$-t$
if(is.na(fit$ssq[expert, "mirrorlogt"])){
equationMirrorLogStudentT <- "Distribution not fitted."
samplingMirrorLogStudentT <- NULL
samplingMirrorLogStudentTText <- NULL
shiftMirrorLogStudentT <- NULL
}else{
m <- signif(fit$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$ssq[expert, "mirrorlogt"])){
plotfit(fit, d = "mirrorlogt")
}
r paste(equationMirrorLogStudentT)
r paste(samplingMirrorLogStudentTText)
r paste(samplingMirrorLogStudentT)
OakleyJ/SHELF documentation built on May 3, 2024, 12:15 a.m.
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if(roulette == TRUE){ bins <- paste0("(",bin.left,", ",bin.right,"]") cat("Distribution elicited using the roulette method.") knitr::kable(data.frame(bins = bins, probs = chips)) }
Elicited cumulative probabilities
```{asis, echo = roulette} Implied cumulative probabilities used for distribution fitting are as follows
```r expert <- 1 sf <- 3 mydf <- data.frame( fit$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$limits[expert, 1]),")")} xMirror <- paste0("(", fit$limits[expert, 2],"-x)") upper <- paste0("`", fit$limits[expert, 2], " - ")
lx <- min(fit$vals[expert, ]) ux <- max(fit$vals[expert, ])
judgementsDF <- data.frame(judgement = c("lower limit", "upper limit", "smallest elicited probability", "largest elicited probability" ) , value = c(fit$limits[expert, "lower"], fit$limits[expert, "upper"], min(fit$probs[expert, ]), max(fit$probs[expert, ]) ) )
```{asis, echo = any(is.na(fit$ssq[expert, ]))} Note: not all distributions have been fitted. The requirements for fitting each distribution are as follows.
-
Normal, Student-t, skew normal: smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Log normal and log Student-t: finite lower limit; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Gamma: finite lower limit; at least one elicited probability between 0 and 1.
-
Beta : finite lower and upper limits; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
-
Mirror gamma: finite upper limit; at least one elicited probability between 0 and 1.
-
Mirror log normal and mirror log Student-t: finite upper limit; smallest elicited probability less than 0.4; largest elicited probability greater than 0.6.
The judgements provided were as follows.
```r knitr::kable(judgementsDF)
Distributions
All parameter values reported to 3 significant figures.
Normal
if(is.na(fit$ssq[expert, "normal"])){ equationNormal <- "Distribution not fitted." samplingNormal <- NULL samplingNormalText <- NULL }else{ mu <- signif(fit$Normal[expert, 1], sf) sigsq <- signif(fit$Normal[expert, 2]^2, sf) equationNormal <- paste0("$$f_X(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp\\left(-\\frac{1}{2 \\sigma^2}(x - \\mu)^2\\right),\\quad -\\infty<x<\\infty,$$ with $$\\mu = ", mu,"$$ $$\\sigma^2 = ", sigsq, "$$") samplingNormal <- paste0("` rnorm(n = 1000, mean = ", mu,", sd = sqrt(", sigsq,"))`" ) samplingNormalText <- "Sample `n = 1000` random values with the command" }
if(!is.na(fit$ssq[expert, "normal"])){ plotfit(fit, d = "normal") }
r paste(equationNormal)
r paste(samplingNormalText)
r paste(samplingNormal)
Student$-t$
if(is.na(fit$ssq[expert, "t"])){ equationT <- "Distribution not fitted." samplingT <- NULL samplingTtext <- NULL }else{ m <- signif(fit$Student.t[expert, 1], 3) s <- signif(fit$Student.t[expert, 2], 3) tdf <- fit$Student.t[expert, 3] equationT <- paste0("$$ f_X(x) = \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\sigma\\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{x - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2},\\quad -\\infty<x<\\infty, $$ with $$\\nu = ", tdf,"$$ $$\\mu = ", m,"$$ $$\\sigma = ", s, "$$") samplingT <- paste0("`", m, " + ", s, " * rt(n = 1000, df = ", tdf, ")`") samplingTtext <- "Sample `n = 1000` random values with the command" }
r paste(equationT)
r paste(samplingTtext)
r paste(samplingT)
Skew normal
if(is.na(fit$ssq[expert, "skewnormal"])){ equationSkewNormal <- "Distribution not fitted." samplingSkewNormal <- NULL samplingSkewNormalText <- NULL }else{ xi <- signif(fit$Skewnormal[expert, 1], sf) omega <- signif(fit$Skewnormal[expert, 2], sf) alpha <- signif(fit$Skewnormal[expert, 3], sf) equationSkewNormal <- paste0("$$f_X(x)=\\frac{2}{\\omega}\\phi\\left(\\frac{x-\\xi}{\\omega}\\right)\\Phi\\left(\\alpha\\left(\\frac{x-\\xi}{\\omega}\\right)\\right),\\quad -\\infty<x<\\infty,$$ where $\\phi(.)$ and $\\Phi(.)$ are the probability density function and cumulative distribution function respectively of the standard normal distribution, and with $$\\xi = ", xi,"$$ $$\\omega = ", omega, "$$ $$\\alpha = ", alpha, "$$") samplingSkewNormal <- paste0("` sn::rsn(n = 1000, xi = ", xi,", omega = ", omega, ", alpha = ",alpha,"))`" ) samplingSkewNormalText <- "Sample `n = 1000` random values with the command" }
if(!is.na(fit$ssq[expert, "skewnormal"])){ plotfit(fit, d = "skewnormal") }
r paste(equationSkewNormal)
r paste(samplingSkewNormalText)
r paste(samplingSkewNormal)
Log normal
if(is.na(fit$ssq[expert, "lognormal"])){ equationLogNormal <- "Distribution not fitted." 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, "lognormal"])){ plotfit(fit, d = "lognormal") }
r paste(equationLogNormal)
r paste(samplingLogNormalText)
r paste(samplingLogNormal)
Gamma
if(is.na(fit$ssq[expert, "gamma"])){ equationGamma <- "Distribution not fitted." samplingGamma <- NULL samplingGammaText <- NULL shiftGamma <- NULL }else{ shape <- signif(fit$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, "gamma"])){ plotfit(fit, d = "gamma") }
r paste(equationGamma)
r paste(samplingGammaText)
r paste(samplingGamma)
Log Student$-t$
if(is.na(fit$ssq[expert, "logt"])){ equationLogStudentT <- "Distribution not fitted." samplingLogStudentT <- NULL samplingLogStudentTText <- NULL shiftLogStudentT <- NULL }else{ m <- signif(fit$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, 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, "beta"])){ scalingBeta <- NULL equationBeta <- "Distribution not fitted." 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$ssq[expert, "beta"])){ plotfit(fit, d = "beta") }
r paste(equationBeta)
r paste(samplingBetaText)
r paste(samplingBeta)
Mirror gamma
if(is.na(fit$ssq[expert, "mirrorgamma"])){ equationMirrorGamma <- "Distribution not fitted." 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$ssq[expert, "mirrorgamma"])){ plotfit(fit, d = "mirrorgamma") }
r paste(equationMirrorGamma)
r paste(samplingMirrorGammaText)
r paste(samplingMirrorGamma)
Mirror Log normal
if(is.na(fit$ssq[expert, "mirrorlognormal"])){ equationMirrorLogNormal <- "Distribution not fitted." 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$ssq[expert, "mirrorlognormal"])){ plotfit(fit, d = "mirrorlognormal") }
r paste(equationMirrorLogNormal)
r paste(samplingMirrorLogNormalText)
r paste(samplingMirrorLogNormal)
Mirror log Student$-t$
if(is.na(fit$ssq[expert, "mirrorlogt"])){ equationMirrorLogStudentT <- "Distribution not fitted." samplingMirrorLogStudentT <- NULL samplingMirrorLogStudentTText <- NULL shiftMirrorLogStudentT <- NULL }else{ m <- signif(fit$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, d = "mirrorlogt") }
r paste(equationMirrorLogStudentT)
r paste(samplingMirrorLogStudentTText)
r paste(samplingMirrorLogStudentT)
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