knitr::opts_chunk$set(echo=FALSE, warning=FALSE, message=FALSE, fig.pos = 'h', fig.align = 'center', fig.height = 3, fig.width = 4)
l <- params$fit$limits[, 1] u <- params$fit$limits[, 2] expert <- 1 sf <- 3 mydf <- data.frame(params$fit$probs[expert, ], t(params$fit$vals) ) colnames(mydf)[1] <- "quantiles" rownames(mydf) <- NULL if(params$entry == "Quantiles"){ subsection = "" knitr::kable(mydf)} if(params$entry == "Roulette"){ subsection = "Implied cumulative probabilities:" knitr::kable(params$chips) } expertnames <- rownames(params$fit$probs)
r subsection
if(params$entry == "Roulette"){ mydf <- data.frame(values = params$fit$vals[1, ], t(params$fit$probs)) knitr::kable(mydf) }
if(params$fit$limits[expert, 1] == 0){ x <- paste0("x") lower <- "`"}else{ lower <- paste0("`", params$fit$limits[expert, 1], " + ")} if(params$fit$limits[expert, 1] > 0){ x <- paste0("(x-", params$fit$limits[expert, 1],")")} if(params$fit$limits[expert, 1] < 0){ x <- paste0("(x+", abs(params$fit$limits[expert, 1]),")")}
lx <- min(params$fit$vals[expert, ]) ux <- max(params$fit$vals[expert, ])
All parameter values reported to 3 significant figures.
plotfit(params$fit, d = "normal")
mu <- signif(params$fit$Normal[, 1], sf) sigsq <- signif(params$fit$Normal[, 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
normaldf <- data.frame(mu, sigsq) colnames(normaldf) <- c("$\\mu$", "$\\sigma^2$") rownames(normaldf) <- expertnames knitr::kable(normaldf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
rnorm(n = 1000, mean =
r mu[1], sd = sqrt(
r sigsq[1]))
m <- signif(params$fit$Student.t[, 1], 3) s <- signif(params$fit$Student.t[, 2], 3) tdf <- params$fit$Student.t[, 3]
plotfit(params$fit, 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
tdataf <- data.frame(m, s, tdf) colnames(tdataf) <- c("$\\mu$", "$\\sigma$", "$\\nu$") rownames(tdataf) <- expertnames knitr::kable(tdataf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r m[1]
+
r s[1]* rt(n = 1000, df =
r tdf[1])
plotfit(params$fit, d = "skewnormal")
xi <- signif(params$fit$Skewnormal[, 1], sf) omega <- signif(params$fit$Skewnormal[, 2], sf) alpha <- signif(params$fit$Skewnormal[, 3], sf)
$$f_{X}(x)=\frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha\left(\frac{x-\xi}{\omega}\right)\right),\quad -\infty<x<\infty,$$ where $\phi(.)$ and $\Phi(.)$ are the probability density function and cumulative distribution function respectively of the standard normal distribution, and with
skewnormaldf <- data.frame(xi, omega, alpha) colnames(skewnormaldf) <- c("$\\xi$", "$\\omega$", "$\\alpha$") rownames(skewnormaldf) <- expertnames knitr::kable(skewnormaldf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
sn::rsn(n = 1000, xi =
r xi[1], omega =
r omega[1], alpha =
r alpha[1])
plotfit(params$fit, d = "lognormal")
$$f_X(x) = \frac{1}{x-L} \times \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2\sigma^2}(\ln (x-L) - \mu)^2\right), \quad x >L,$$
and $f_X(x)=0$ otherwise, with
mu <- signif(params$fit$Log.normal[, 1], 3) sigsq <- signif(params$fit$Log.normal[, 2]^2, 3) lnormdf <- data.frame(mu, sigsq, l) colnames(lnormdf) <- c("$\\mu$", "$\\sigma^2$", "$L$") rownames(lnormdf) <- expertnames knitr::kable(lnormdf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r l[1]
+ rlnorm(n = 1000, meanlog =
r mu[1], sdlog =
r signif(sqrt(sigsq[1]), 3))
plotfit(params$fit, d = "gamma")
$$f_X(x) =\frac{\beta ^ {\alpha}}{\Gamma(\alpha)}(x-L)^{\alpha - 1} \exp\left(- \beta (x-L)\right), \quad x >L,$$ and $f_X(x)=0$ otherwise, with
shape <- signif(params$fit$Gamma[, 1], 3) rate <- signif(params$fit$Gamma[, 2], 3) gammadf <- data.frame(shape, rate, l) colnames(gammadf) <- c("$\\alpha$", "$\\beta$", "$L$") rownames(gammadf) <- expertnames knitr::kable(gammadf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r l[1]
+ rgamma(n = 1000, shape =
r shape[1], rate =
r rate[1])
plotfit(params$fit, d = "logt")
$$f_X(x) =\frac{1}{x-L} \times \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\times\sigma\times \sqrt{\nu \pi}} \left(1 + \frac{1}{\nu}\left(\frac{\ln (x-L) - \mu}{\sigma}\right)^2\right)^{-(\nu + 1)/2}, \quad x >L,$$ and $f_X(x)=0$ otherwise, with
m <- signif(params$fit$Log.Student.t[, 1], 3) s <- signif(params$fit$Log.Student.t[, 2], 3) tdf <- params$fit$Log.Student.t[, 3] logtdf <- data.frame(m, s, tdf, l) colnames(logtdf) <- c("$\\mu$", "$\\sigma$", "$\\nu$", "$L$") rownames(logtdf) <- expertnames knitr::kable(logtdf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r l[1]
+ exp(
r m[1]+
r s[1]* rt(n = 1000, df =
r tdf[1]))
plotfit(params$fit, d = "beta")
$$f_X(x) = \frac{1}{U-L}\times\frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \left(\frac{x-L}{U-L}\right) ^{\alpha - 1} \left(1 - \left(\frac{x-L}{U-L}\right)\right)^{\beta - 1}, \quad L < x <U,$$ and $f_X(x) = 0$ otherwise, with
shape1 <- signif(params$fit$Beta[, 1], 3) shape2 <- signif(params$fit$Beta[, 2], 3) betadf <- data.frame(shape1, shape2, l, params$fit$limits[, 2]) colnames(betadf) <- c("$\\alpha$", "$\\beta$", "$L$", "$U$") rownames(betadf) <- expertnames knitr::kable(betadf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r l[1]
+ (
r params$fit$limits[1, 2] - l[1] ) * rbeta(n = 1000, shape1 =
r shape1[1], shape2 =
r shape2[1])
plotfit(params$fit, d = "mirrorlognormal")
$$f_X(x) = \frac{1}{U-x} \times \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2\sigma^2}(\ln (U-x) - \mu)^2\right), \quad x <U,$$
and $f_X(x)=0$ otherwise, with
mu <- signif(params$fit$mirrorlognormal[, 1], 3) sigsq <- signif(params$fit$mirrorlognormal[, 2]^2, 3) lnormdf <- data.frame(mu, sigsq, u) colnames(lnormdf) <- c("$\\mu$", "$\\sigma^2$", "$U$") rownames(lnormdf) <- expertnames knitr::kable(lnormdf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r u[1]
- rlnorm(n = 1000, meanlog =
r mu[1], sdlog =
r signif(sqrt(sigsq[1]), 3))
plotfit(params$fit, d = "mirrorgamma")
$$f_X(x) =\frac{\beta ^ {\alpha}}{\Gamma(\alpha)}(U-x)^{\alpha - 1} \exp\left(- \beta (U-x)\right), \quad x <U,$$ and $f_X(x)=0$ otherwise, with
shape <- signif(params$fit$mirrorgamma[, 1], 3) rate <- signif(params$fit$mirrorgamma[, 2], 3) gammadf <- data.frame(shape, rate, u) colnames(gammadf) <- c("$\\alpha$", "$\\beta$", "$U$") rownames(gammadf) <- expertnames knitr::kable(gammadf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r u[1]
- rgamma(n = 1000, shape =
r shape[1], rate =
r rate[1])
plotfit(params$fit, d = "mirrorlogt")
$$f_X(x) =\frac{1}{U-x} \times \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\times\sigma\times \sqrt{\nu \pi}} \left(1 + \frac{1}{\nu}\left(\frac{\ln (U-x) - \mu}{\sigma}\right)^2\right)^{-(\nu + 1)/2}, \quad x <U,$$ and $f_X(x)=0$ otherwise, with
m <- signif(params$fit$mirrorlogt[, 1], 3) s <- signif(params$fit$mirrorlogt[, 2], 3) tdf <- params$fit$mirrorlogt[, 3] logtdf <- data.frame(m, s, tdf, u) colnames(logtdf) <- c("$\\mu$", "$\\sigma$", "$\\nu$", "$U$") rownames(logtdf) <- expertnames knitr::kable(logtdf, escape = FALSE)
Sample n = 1000
random values (for expert r expertnames[1]
) with the command
r u[1]
- exp(
r m[1]+
r s[1]* rt(n = 1000, df =
r tdf[1]))
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