---
title: "Fitted distributions (extension method)"
output:
  html_document: default
  pdf_document: default
  word_document: default
date: "`r format(Sys.time(), '%d %B %Y, %H:%M')`"
fontsize: 11pt
params:
  fit1: NA
  fit2: NA
  cp: NA
  d: NA
  m1: NA
  m2: NA
  yLimits: NA
  link: NA
  yCP: NA
  xMed: NA
  df1: NA
  
---

# Marginal distribution of $Y$

```{r global_options, include=FALSE}
knitr::opts_chunk$set(echo=FALSE, warning=FALSE, message=FALSE)
showplots <- FALSE
```

### Elicited cumulative probabilities

```{r}
expert <- 1
sf <- 3
mydf <- data.frame( params$fit1$vals[expert, ], params$fit1$probs[expert, ])
colnames(mydf) <- c( "$x$", "$P(X\\le x)$")
knitr::kable(mydf)
```


```{r}
if(params$fit1$limits[expert, 1] == 0){
  x <- paste0("x")
  lower <- "`"}else{
    lower <- paste0("`", params$fit1$limits[expert, 1], " + ")}

if(params$fit1$limits[expert, 1] > 0){
  x <- paste0("(x-", params$fit1$limits[expert, 1],")")}
if(params$fit1$limits[expert, 1] < 0){
  x <- paste0("(x+", abs(params$fit1$limits[expert, 1]),")")}
```


```{r}
lx <- min(params$fit1$vals[expert, ])
ux <- max(params$fit1$vals[expert, ])
```


### Distributions

All parameter values reported to 3 significant figures.

#### Normal 
```{r, echo = FALSE}
mu <- signif(params$fit1$Normal[expert, 1], sf)
sigsq <- signif(params$fit1$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`)) `

<hr>

#### Student$-t$ 
```{r, echo = FALSE}
m <- signif(params$fit1$Student.t[expert, 1], 3)
s <- signif(params$fit1$Student.t[expert, 2], 3)
tdf <- params$fit1$Student.t[expert, 3]
```

$$
f_X(x) = \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\sigma\sqrt{\nu \pi}} \left(1 + \frac{1}{\nu}\left(\frac{x - \mu}{\sigma}\right)^2\right)^{-(\nu + 1)/2},\quad -\infty<x<\infty,
$$
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`) `
<hr>
#### Log normal 


```{r, echo = FALSE}
if(is.na(params$fit1$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(params$fit1$Log.normal[expert, 1], 3)
  sigsq <- signif(params$fit1$Log.normal[expert, 2]^2, 3)  
  
  equationLogNormal <- paste0("$$f_X(x) = \\frac{1}{", x, "} \\times \\frac{1}{\\sqrt{2\\pi\\sigma^2}} 
\\exp\\left(-\\frac{1}{2\\sigma^2}(\\ln ", x,"  - \\mu)^2\\right), \\quad x >", params$fit1$limits[expert, 1],"$$ 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(params$fit1$limits[expert, 1] != 0){
    shiftLognormal <- paste0("Log normal distribution shifted to have support over the interval $(",
                               params$fit1$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLognormal <- NULL
  }
  
} 
```




`r paste(shiftLognormal)`

`r paste(equationLogNormal)`


`r paste(samplingLogNormalText)`

`r paste(samplingLogNormal)`


<hr>
#### Gamma  

```{r}
if(is.na(params$fit1$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(params$fit1$Gamma[expert, 1], 3)
  rate <- signif(params$fit1$Gamma[expert, 2], 3)  
  
  equationGamma <- paste0("$$f_X(x) =\\frac{\\beta ^ {\\alpha}}{\\Gamma(\\alpha)}",  x,
"^{\\alpha - 1} \\exp\\left(- \\beta ", x,"\\right), \\quad x >", params$fit1$limits[expert, 1],"$$ 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(params$fit1$limits[expert, 1] != 0){
    shiftGamma <- paste0("Gamma distribution shifted to have support over the interval $(",
                               params$fit1$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftGamma <- NULL
  }
  
} 
```

`r paste(shiftGamma)`

`r paste(equationGamma)`

`r paste(samplingGammaText)`

`r paste(samplingGamma)`

<hr>
#### Log Student$-t$ 

```{r}
if(is.na(params$fit1$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(params$fit1$Log.Student.t[expert, 1], 3)
  s <- signif(params$fit1$Log.Student.t[expert, 2], 3)
  tdf <- params$fit1$Log.Student.t[expert, 3]
  
  equationLogStudentT <- paste0("$$f_X(x) =\\frac{1}{", x, "} \\times \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\times\\sigma\\times \\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{\\ln ", x, " - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2}, \\quad x >", params$fit1$limits[expert, 1],"$$ 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(params$fit1$limits[expert, 1] != 0){
    shiftLogStudentT <- paste0("Log Student-$t$ distribution shifted to have support over the interval $(",
                               params$fit1$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLogStudentT <- NULL
  }
} 
```

`r paste(shiftLogStudentT)`

`r paste(equationLogStudentT)`



`r paste(samplingLogStudentTText)`

`r paste(samplingLogStudentT)`



<hr>
#### Beta 
```{r, echo = FALSE}
if(params$fit1$limits[expert, 1] == 0)
  x <- paste0("x")
if(params$fit1$limits[expert, 1] > 0)
  x <- paste0("x-", params$fit1$limits[1, expert])
if(params$fit1$limits[expert, 1] < 0)
  x <- paste0("x+", abs(params$fit1$limits[1, expert]))
if(params$fit1$limits[expert, 2] < Inf & params$fit1$limits[expert, 1] > -Inf){
  r <- params$fit1$limits[expert, 2] - params$fit1$limits[expert, 1]
  if(r !=1){
  mult = paste0(r, " *")}else{
    mult = ""
  }
}


shape1 <- signif(params$fit1$Beta[expert, 1], 3)
shape2 <- signif(params$fit1$Beta[expert, 2], 3)

```

```{r, echo = FALSE}
if(is.na(params$fit1$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 & params$fit1$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 & params$fit1$limits[expert, 1] != 0){
    equationBeta <- paste0("$$f_X(x) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(", x,"\\right) ^{\\alpha - 1} \\left(1 - \\left(", x,"\\right)\\right)^{\\beta - 1}, \\quad  ", params$fit1$limits[expert, 1]," < x <", params$fit1$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          params$fit1$limits[expert, 1],
                          ", ",
                          params$fit1$limits[expert, 2],
                          "].")
  }
  
  
  if(r !=1){
    equationBeta <- paste0("$$f_X(x) = \\frac{1}{", r, "}\\times\\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(\\frac{", x,"}{", r,"}\\right) ^{\\alpha - 1} \\left(1 - \\left(\\frac{", x,"}{", r,"}\\right)\\right)^{\\beta - 1}, \\quad  ", params$fit1$limits[expert, 1]," < x <", params$fit1$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          params$fit1$limits[expert, 1],
                          ", ",
                          params$fit1$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)`

`r paste(equationBeta)`


`r paste(samplingBetaText)`

`r paste(samplingBeta)`








# Conditional distribution of $X$, given $Y$ takes its median value



### Elicited cumulative probabilities

```{r}
expert <- 1
sf <- 3
mydf <- data.frame( params$fit2$vals[expert, ], params$fit2$probs[expert, ])
colnames(mydf) <- c( "$x$", "$P(X\\le x)$")
knitr::kable(mydf)
```


```{r}
if(params$fit2$limits[expert, 1] == 0){
  x <- paste0("x")
  lower <- "`"}else{
    lower <- paste0("`", params$fit2$limits[expert, 1], " + ")}

if(params$fit2$limits[expert, 1] > 0){
  x <- paste0("(x-", params$fit2$limits[expert, 1],")")}
if(params$fit2$limits[expert, 1] < 0){
  x <- paste0("(x+", abs(params$fit2$limits[expert, 1]),")")}
```


```{r}
lx <- min(params$fit2$vals[expert, ])
ux <- max(params$fit2$vals[expert, ])
```


### Distributions

All parameter values reported to 3 significant figures.

#### Normal 
```{r, echo = FALSE}
mu <- signif(params$fit2$Normal[expert, 1], sf)
sigsq <- signif(params$fit2$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`)) `

<hr>

#### Student$-t$ 
```{r, echo = FALSE}
m <- signif(params$fit2$Student.t[expert, 1], 3)
s <- signif(params$fit2$Student.t[expert, 2], 3)
tdf <- params$fit2$Student.t[expert, 3]
```

$$
f_X(x) = \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\sigma\sqrt{\nu \pi}} \left(1 + \frac{1}{\nu}\left(\frac{x - \mu}{\sigma}\right)^2\right)^{-(\nu + 1)/2},\quad -\infty<x<\infty,
$$
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`) `
<hr>
#### Log normal 


```{r, echo = FALSE}
if(is.na(params$fit2$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(params$fit2$Log.normal[expert, 1], 3)
  sigsq <- signif(params$fit2$Log.normal[expert, 2]^2, 3)  
  
  equationLogNormal <- paste0("$$f_X(x) = \\frac{1}{", x, "} \\times \\frac{1}{\\sqrt{2\\pi\\sigma^2}} 
\\exp\\left(-\\frac{1}{2\\sigma^2}(\\ln ", x,"  - \\mu)^2\\right), \\quad x >", params$fit2$limits[expert, 1],"$$ 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(params$fit2$limits[expert, 1] != 0){
    shiftLognormal <- paste0("Log normal distribution shifted to have support over the interval $(",
                               params$fit2$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLognormal <- NULL
  }
  
} 
```




`r paste(shiftLognormal)`

`r paste(equationLogNormal)`


`r paste(samplingLogNormalText)`

`r paste(samplingLogNormal)`


<hr>
#### Gamma  

```{r}
if(is.na(params$fit2$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(params$fit2$Gamma[expert, 1], 3)
  rate <- signif(params$fit2$Gamma[expert, 2], 3)  
  
  equationGamma <- paste0("$$f_X(x) =\\frac{\\beta ^ {\\alpha}}{\\Gamma(\\alpha)}",  x,
"^{\\alpha - 1} \\exp\\left(- \\beta ", x,"\\right), \\quad x >", params$fit2$limits[expert, 1],"$$ 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(params$fit2$limits[expert, 1] != 0){
    shiftGamma <- paste0("Gamma distribution shifted to have support over the interval $(",
                               params$fit2$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftGamma <- NULL
  }
  
} 
```

`r paste(shiftGamma)`

`r paste(equationGamma)`

`r paste(samplingGammaText)`

`r paste(samplingGamma)`

<hr>
#### Log Student$-t$ 

```{r}
if(is.na(params$fit2$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(params$fit2$Log.Student.t[expert, 1], 3)
  s <- signif(params$fit2$Log.Student.t[expert, 2], 3)
  tdf <- params$fit2$Log.Student.t[expert, 3]
  
  equationLogStudentT <- paste0("$$f_X(x) =\\frac{1}{", x, "} \\times \\frac{\\Gamma((\\nu + 1)/2)}{\\Gamma(\\nu/2)\\times\\sigma\\times \\sqrt{\\nu \\pi}} \\left(1 + \\frac{1}{\\nu}\\left(\\frac{\\ln ", x, " - \\mu}{\\sigma}\\right)^2\\right)^{-(\\nu + 1)/2}, \\quad x >", params$fit2$limits[expert, 1],"$$ 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(params$fit2$limits[expert, 1] != 0){
    shiftLogStudentT <- paste0("Log Student-$t$ distribution shifted to have support over the interval $(",
                               params$fit2$limits[expert, 1],
                               ",\\, \\infty )$.")
  }else{
    shiftLogStudentT <- NULL
  }
} 
```

`r paste(shiftLogStudentT)`

`r paste(equationLogStudentT)`



`r paste(samplingLogStudentTText)`

`r paste(samplingLogStudentT)`



<hr>
#### Beta 
```{r, echo = FALSE}
if(params$fit2$limits[expert, 1] == 0)
  x <- paste0("x")
if(params$fit2$limits[expert, 1] > 0)
  x <- paste0("x-", params$fit2$limits[1, expert])
if(params$fit2$limits[expert, 1] < 0)
  x <- paste0("x+", abs(params$fit2$limits[1, expert]))
if(params$fit2$limits[expert, 2] < Inf & params$fit2$limits[expert, 1] > -Inf){
  r <- params$fit2$limits[expert, 2] - params$fit2$limits[expert, 1]
  if(r !=1){
  mult = paste0(r, " *")}else{
    mult = ""
  }
}


shape1 <- signif(params$fit2$Beta[expert, 1], 3)
shape2 <- signif(params$fit2$Beta[expert, 2], 3)

```

```{r, echo = FALSE}
if(is.na(params$fit2$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 & params$fit2$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 & params$fit2$limits[expert, 1] != 0){
    equationBeta <- paste0("$$f_X(x) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(", x,"\\right) ^{\\alpha - 1} \\left(1 - \\left(", x,"\\right)\\right)^{\\beta - 1}, \\quad  ", params$fit2$limits[expert, 1]," < x <", params$fit2$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          params$fit2$limits[expert, 1],
                          ", ",
                          params$fit2$limits[expert, 2],
                          "].")
  }
  
  
  if(r !=1){
    equationBeta <- paste0("$$f_X(x) = \\frac{1}{", r, "}\\times\\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\left(\\frac{", x,"}{", r,"}\\right) ^{\\alpha - 1} \\left(1 - \\left(\\frac{", x,"}{", r,"}\\right)\\right)^{\\beta - 1}, \\quad  ", params$fit2$limits[expert, 1]," < x <", params$fit2$limits[expert, 2],",$$ and $f_X(x) = 0$ otherwise, with $$\\alpha = ", shape1, ",$$ $$ \\beta = ", shape2, ".$$")
    scalingBeta <- paste0("Fitted beta distribution is scaled to the interval [",
                          params$fit2$limits[expert, 1],
                          ", ",
                          params$fit2$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)`

`r paste(equationBeta)`


`r paste(samplingBetaText)`

`r paste(samplingBeta)`



# Median function

```{r, echo = FALSE, fig.align = "center"}
plotConditionalMedianFunction(yCP = params$yCP, xMed = params$xMed, 
                              yLimits = params$yLimits,
                              link = params$link)
```


# Marginal distribution of X

```{r}
d1 <- switch(params$d[1],
                       "normal" = "normal",
                       "t" = "Student-t",
                       "gamma" = "gamma",
                       "lognormal" = "log normal",
                       "logt" = "log Student-t",
                       "beta" = "beta",
                       "hist" = "histogram",
                       "best" = as.character(params$fit1$best.fitting[1, 1]))

d2 <- switch(params$d[2],
                       "normal" = "normal",
                       "t" = "Student-t",
                       "gamma" = "gamma",
                       "lognormal" = "log normal",
                       "logt" = "log Student-t",
                       "beta" = "beta",
                       "hist" = "histogram",
                       "best" = as.character(params$fit2$best.fitting[1, 1]))
```



Marginal distribution of $X$, obtained using a `r paste(d1)` distribution for $Y$ and a `r paste(d2)` distribution for $X|Y$:

```{r, echo = FALSE, fig.align = "center", message = FALSE}
library(ggplot2)

ggplot(params$df1, aes(x = X, y = ..density..))+
        geom_density(fill = "steelblue")


```

