Package 'rriskDistributions'

Fitting distributions to given data or known quantiles

Description

This packages provides a collection of functions for estimation parameters of continuous or discrete distributions (related to the rrisk project) to given data or to known quantiles.

Details

This package is a part of the rrisk project and contains functions for fitting distributions to given data or by known quantiles. This package does not depend on the whole rrisk project and can be used separately. The rrisk project can be downloaded from http://www.bfr.bund.de/cd/52158.

The main functions fit.perc and fit.cont call a GUI that allows users to choose an appropriate distribution family to given data or to known quantiles without any knowledge of the R syntax.

Note

Fitting by given quantiles: a typical application is the definition of a distribution based on expert opinion on some quantiles (e.g., the 2.5th, median and 97.5th) of the trial to be modelled. rrisk has a functionality, to fit all continuous or discrete distributions simultaneously without urging the user to specify the distribution family in advance.

Author(s)

Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting),
Matthias Greiner [email protected] (Federal Institute for Risk Assessment, Germany),
Kristin Tolksdorf [email protected] (Federal Institute for Risk Assessment, Germany),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting)

Examples

q <- stats::qweibull(p = c(0.025, 0.5, 0.975), shape = 2, scale = 3)
get.weibull.par(q = q)
q <- stats::qweibull(p = c(0.025, 0.5, 0.975), shape = 0.01, scale = 1)
get.weibull.par(q = q)

p <- c(0.025, 0.50, 0.975)
q <- c(9.68, 29.2, 50.98)
fit.results <- rriskFitdist.perc(p, q, show.output = FALSE)
plotDiagnostics.perc(fit.results)

p <- c(0.25, 0.50, 0.75)
q <- c(9.68, 29.2, 50.98)
fit.results <- rriskFitdist.perc(p, q, show.output = FALSE)
plotDiagnostics.perc(fit.results)
plotDiagnostics.perc(fit.results, tolPlot = 2)

## Not run: 
  if( class(tcltk::tclRequire("Tktable")) == "tclObj" ) {
    res.fitcont <- fit.cont(data2fit = rnorm(100))
    res.fitcont
  }
  if( class(tcltk::tclRequire("Tktable")) == "tclObj" ) {
    res.fitperc <- fit.perc()
    res.fitperc
  }

## End(Not run)

---

GUI for fitting continuous distributions to given data

Description

This function provides a GUI for choosing a most appropriate continuous distribution fitted to given data.

Usage

fit.cont(data2fit)

Arguments

data2fit

numerical vector, data to be fitted.

Value

Returns chosen continuous distribution, estimated parameters and data, on which the fitting is based.

Note

This function is used for defining a Monte-Carlo random variate item (mcrv) in the rrisk project.

Author(s)

Matthias Greiner [email protected] (BfR),
Kristin Tolksdorf [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

Examples

## Not run: 
  if ( class(tcltk::tclRequire("Tktable")) == "tclObj" ) {
    res1 <- fit.cont(data2fit = rgamma(374, 4, 0.08))
    res1

    res2 <- fit.cont(data2fit = rbeta(300, shape1 = 1, shape2 = 2))
    res2

    res3 <- fit.cont(data2fit = mc2d::rtriang(300, min = 1, mode = 3, max = 10))
    res3

    res4 <- fit.cont(data2fit = stats::rnorm(300))
    res4
  }

## End(Not run)

---

Choosing distribution by given quantiles

Description

This function provides a GUI for choosing a most appropriate continuous distribution for known quantiles.

Usage

fit.perc(p = c(0.025, 0.5, 0.975), q = stats::qnorm(p), show.output = FALSE,
  tolPlot = 0.1, tolConv = 0.001, fit.weights = rep(1, length(p)))

Arguments

p	numerical vector of probabilities.
q	numerical vector of quantiles.
show.output	logical, if TRUE the output of the fitting functions get.distribution.par will be shown.
tolPlot	single positive numerical value giving a tolerance for plotting graphical diagnostics. If the sums of the differences between the distribution percentiles and the given percentiles are smaller than this value, the distribution will be plotted.
tolConv	positive numerical value, the absolute convergence tolerance for reaching zero.
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.

Details

The argument tolPlot defines a tolerance for plotting graphical diagnostics. If the sums of the differences between the percentiles of the estimated distribution and the given percentiles are smaller than this value, the distribution will be plotted.

The items of the probability vector p should lie between 0 and 1.

Value

Returns a named list containing a chosen distribution, its estimated parameters and the data on which the estimation is based.

Note

This function is used for defining a Monte-Carlo random variate item (mcrv) in the rrisk project.

Author(s)

Matthias Greiner [email protected] (BfR),
Kristin Tolksdorf [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

Examples

## Not run: 
    chosenDistr1 <- fit.perc()
    chosenDistr1
    
    chosenDistr2 <- fit.perc(tolPlot = 5)
    chosenDistr2
    
    chosenDistr3 <- fit.perc(p = c(0.3, 0.8, 0.9), q = c(10, 20, 40))
    chosenDistr3
    
    chosenDistr4 <- fit.perc(p = c(0.3, 0.8, 0.9), q = c(10, 30, 40))
    chosenDistr4
    
    chosenDistr5 <- fit.perc(p = c(0.3, 0.8, 0.9), q = c(10, 30, 40), tolPlot = 10)
    chosenDistr5

    ## Fitting a PERT distribution
    p <- c(0.025, 0.5, 0.6, 0.975)
    q <- round(mc2d::qpert(p = p, min = 0, mode = 3, max = 10, shape = 5), digits = 2)
    chosenDistr6 <- fit.perc(p = p, q = q, tolPlot = 10)
    chosenDistr6

## End(Not run)

---

Fitting parameters of a Beta distribution from two or more quantiles

Description

get.beta.par returns the parameters of estimated beta distribution where the pth percentiles match with the quantiles q.

Usage

get.beta.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE, plot = TRUE,
     tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the numbers of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.beta.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method CG will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a Beta distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the eror massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good til very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pbeta for distribution implementation details.

Examples

q <- stats::qbeta(p = c(0.025, 0.5, 0.975), shape1 = 2, shape2 = 3)
old.par <- graphics::par(mfrow = c(2, 3))
get.beta.par(q = q)
get.beta.par(q = q, scaleX = c(0.001, 0.999))
get.beta.par(q = q, fit.weights = c(10, 1, 10))
get.beta.par(q = q, fit.weights = c(1, 10, 1))
get.beta.par(q = q, fit.weights = c(100, 1, 100))
get.beta.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qbeta(p = c(0.025, 0.5, 0.975), shape1 = 1, shape2 = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.beta.par(q = q)
get.beta.par(q = q, fit.weights = c(10, 1, 10))
get.beta.par(q = q, fit.weights = c(1, 10, 1))
get.beta.par(q = q, fit.weights = c(100, 1, 100))
get.beta.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qbeta(p = c(0.025, 0.5, 0.975), shape1 = 0.3, shape2 = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.beta.par(q = q)
get.beta.par(q = q, fit.weights = c(10, 1, 10))
get.beta.par(q = q, fit.weights = c(1, 10, 1))
get.beta.par(q = q, fit.weights = c(100, 1, 100))
get.beta.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qbeta(p = c(0.025, 0.975), shape1 = 2, shape2 = 3)
old.par <- graphics::par(mfrow = c(2, 3))
get.beta.par(p = c(0.025, 0.975), q = q)
get.beta.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.beta.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
get.beta.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.beta.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
graphics::par(old.par)

---

Fitting parameters of a Cauchy distribution from two or more quantiles

Description

get.cauchy.par returns the parameters of a Cauchy distribution where the pth percentiles match with the quantiles q.

Usage

get.cauchy.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.cauchy.par uses the R function optim with the method L-BFGS-B.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a Cauchy distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the eror massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good til very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pcauchy for distribution implementation details.

Examples

q <- stats::qcauchy(p = c(0.025, 0.5, 0.975), location = 0, scale = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.cauchy.par(q = q)
get.cauchy.par(q = q, scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.5, 0.5))
graphics::par(old.par)

q <- stats::qcauchy(p = c(0.025, 0.5, 0.975), location = 3, scale = 5)
old.par <- graphics::par(mfrow = c(2, 3))
get.cauchy.par(q = q, scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.5, 0.5))
graphics::par(old.par)

q <- stats::qcauchy(p = c(0.025, 0.5, 0.975), location = 0.1, scale = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.cauchy.par(q = q, scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.5, 0.5))
get.cauchy.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.5, 0.5))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qcauchy(p = c(0.025, 0.975), location = 0.1, scale = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.cauchy.par(p = c(0.025, 0.975), q = q, scaleX = c(0.5, 0.5))
get.cauchy.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1), scaleX = c(0.5, 0.5))
get.cauchy.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10), scaleX = c(0.5, 0.5))
get.cauchy.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100), scaleX = c(0.5, 0.5))
graphics::par(old.par)

---

Fitting parameter of a chi-square distribution from one or more quantiles

Description

get.chisq.par returns the parameters of a chi-square distribution where the pth percentiles match with the quantiles q.

Usage

get.chisq.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)). for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE)

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least one. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.chisq.par uses the R function optim with the method L-BFGS-B.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted, if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values. The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a chi-square distribution or missing values (NA's), if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pchisq for distribution implementation details.

Examples

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisq.par(q = q)
get.chisq.par(q = q, fit.weights = c(10, 1, 10))
get.chisq.par(q = q, fit.weights = c(100, 1, 100))
get.chisq.par(q = q, fit.weights = c(1, 10, 1))
get.chisq.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisq.par(q = q, scaleX = c(0.1, 0.1))
get.chisq.par(q = q, fit.weights = c(10, 1, 10))
get.chisq.par(q = q, fit.weights = c(100, 1, 100))
get.chisq.par(q = q, fit.weights = c(1, 10, 1))
get.chisq.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 20)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisq.par(q = q)
get.chisq.par(q = q, fit.weights = c(10, 1, 10))
get.chisq.par(q = q, fit.weights = c(100, 1, 100))
get.chisq.par(q = q, fit.weights =c(1, 10, 1))
get.chisq.par(q = q, fit.weights =c(1, 100, 1))
graphics::par(old.par)

## example with only one quantile
q <- stats::qchisq(p = c(0.025), df = 20)
old.par <- graphics::par(mfrow = c(1, 3))
get.chisq.par(p = c(0.025), q = q)
get.chisq.par(p = c(0.025), q = q, fit.weights = 10)
get.chisq.par(p = c(0.025), q = q, fit.weights = 100)
graphics::par(old.par)

---

Fitting parameters of a non-central chi-square distribution from one or more quantiles

Description

get.chisqnc.par returns the parameters of a non-central chi-square distribution where the pth percentiles match with the quantiles q.

Usage

get.chisqnc.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weights must be identical and should be at least one. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.chisqnc.par uses the R function optim with the method L-BFGS-B.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a non-central chi-square distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the eror massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pchisq for distribution implementation details.

Examples

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 2, ncp = 4)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisqnc.par(q = q)
get.chisqnc.par(q = q, scaleX = c(0.1, 0.9999999))
get.chisqnc.par(q = q, fit.weights = c(100, 1, 100))
get.chisqnc.par(q = q, fit.weights = c(10, 1, 10))
get.chisqnc.par(q = q, fit.weights = c(1, 100, 1))
get.chisqnc.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 0.1, ncp = 0.4)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisqnc.par(q = q)
get.chisqnc.par(q = q, fit.weights = c(100, 1, 100))
get.chisqnc.par(q = q, fit.weights = c(10, 1, 10))
get.chisqnc.par(q = q, fit.weights = c(1, 100, 1))
get.chisqnc.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qchisq(p = c(0.025, 0.5, 0.975), df = 1, ncp = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisqnc.par(q = q)
get.chisqnc.par(q = q, fit.weights = c(100, 1, 100))
get.chisqnc.par(q = q, fit.weights = c(10, 1, 10))
get.chisqnc.par(q = q, fit.weights = c(1, 100, 1))
get.chisqnc.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantile
q <- stats::qchisq(p = c(0.025, 0.95), df = 20, ncp = 20)
old.par <- graphics::par(mfrow = c(2, 3))
get.chisqnc.par(p = c(0.025, 0.975), q = q)
get.chisqnc.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.chisqnc.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
get.chisqnc.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.chisqnc.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
graphics::par(old.par)

---

Fitting parameters of an exponential distribution from one or more quantiles

Description

get.exp.par returns the parameters of an exponential distribution where the pth percentiles match with the quantiles q.

Usage

get.exp.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least one. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.exp.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values. The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of an exponential distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the eror massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good til very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pexp for distribution implementation details.

Examples

q <- stats::qexp(p = c(0.025, 0.5, 0.975), rate = 2)
old.par <- graphics::par(mfrow = c(2, 3))
get.exp.par(q = q)
get.exp.par(q = q, fit.weights = c(100, 1, 100))
get.exp.par(q = q, fit.weights = c(10, 1, 10))
get.exp.par(q = q, fit.weights = c(1, 100, 1))
get.exp.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qexp(p = c(0.025, 0.5, 0.975), rate = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.exp.par(q = q)
get.exp.par(q = q, fit.weights = c(100, 1, 100))
get.exp.par(q = q, fit.weights = c(10, 1, 10))
get.exp.par(q = q, fit.weights = c(1, 100, 1))
get.exp.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qexp(p = c(0.025, 0.5, 0.975), rate = 0.001)
old.par <- graphics::par(mfrow = c(2, 3))
get.exp.par(q = q)
get.exp.par(q = q, fit.weights = c(100, 1, 100))
get.exp.par(q = q, fit.weights = c(10, 1, 10))
get.exp.par(q = q, fit.weights = c(1, 100, 1))
get.exp.par(q = q, tol = 0.2, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qexp(p = c(0.025, 0.5, 0.975), rate = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.exp.par(q = q)
get.exp.par(q = q, fit.weights = c(100, 1, 100))
get.exp.par(q = q, fit.weights = c(10, 1, 10))
get.exp.par(q = q, fit.weights = c(1, 100, 1))
get.exp.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only one quantile
q <- stats::qexp(p = c(0.025), rate = 2)
old.par <- graphics::par(mfrow = c(1, 3))
get.exp.par(p = c(0.025), q = q)
get.exp.par(p = c(0.025), q = q, fit.weights = 10)
get.exp.par(p = c(0.025), q = q, fit.weights = 100)
graphics::par(old.par)

---

Fitting parameters of a F distribution from two or more quantiles

Description

get.f.par returns the parameters of a F distribution where the pth percentiles match with the quantiles q.

Usage

get.f.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.f.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a F distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See stats::pf for distribution implementation details.

Examples

q <- stats::qf(p = c(0.025, 0.5, 0.975), df1 = 2, df2 = 10)
old.par <- graphics::par(mfrow = c(2, 3))
get.f.par(q = q, scaleX = c(0.1, 0.5))
get.f.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.1, 0.5))
get.f.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.1, 0.5))
get.f.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.1, 0.5))
get.f.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.1, 0.5))
graphics::par(old.par)

q <- stats::qf(p = c(0.025, 0.5, 0.975), df1 = 0.2, df2 = 0.3)
old.par <- graphics::par(mfrow = c(2, 3))
get.f.par(q = q, scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.1, 0.999))
get.f.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.1, 0.9999))
get.f.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.1, 0.9999))
graphics::par(old.par)

q <- stats::qf(p = c(0.025, 0.5, 0.975), df1 = 1, df2 = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.f.par(q = q, scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(100, 1, 100), scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(10, 1, 10), scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.1, 0.2))
get.f.par(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.1, 0.2))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qf(p = c(0.025, 0.975), df1 = 2, df2 = 3)
old.par <- graphics::par(mfrow = c(1, 3))
get.f.par(p = c(0.025, 0.975), q = q)
get.f.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.f.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
graphics::par(old.par)

---

Fitting parameters of a gamma distribution from two or more quantiles

Description

get.gamma.par returns the parameters of a gamma distribution where the pth percentiles match with the quantiles q.

Usage

get.gamma.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)). for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.gamma.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a gamma distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pgamma for distribution implementation details.

Examples

q <- stats::qgamma(p = c(0.025, 0.5, 0.975), shape = 10, rate = 10)
old.par <- graphics::par(mfrow = c(2, 3))
get.gamma.par(q = q)
get.gamma.par(q = q, scaleX = c(0.00001, 0.9999))
get.gamma.par(q = q, fit.weights = c(100, 1, 100))
get.gamma.par(q = q, fit.weights = c(10, 1, 10))
get.gamma.par(q = q, fit.weights = c(1, 100, 1))
get.gamma.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qgamma(p = c(0.025, 0.5, 0.975), shape = 0.1, rate = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.gamma.par(q = q)
get.gamma.par(q = q, fit.weights = c(100, 1, 100))
get.gamma.par(q = q, fit.weights = c(10, 1, 10))
get.gamma.par(q = q, fit.weights = c(1, 100, 1))
get.gamma.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qgamma(p = c(0.025, 0.5, 0.975), shape = 1, rate = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.gamma.par(q = q)
get.gamma.par(q = q, fit.weights = c(100, 1, 100))
get.gamma.par(q = q, fit.weights = c(10, 1, 10))
get.gamma.par(q = q, fit.weights = c(1, 100, 1))
get.gamma.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qgamma(p = c(0.025, 0.975), shape = 10, rate = 10)
old.par <- graphics::par(mfrow = c(2, 3))
get.gamma.par(p = c(0.025, 0.975), q = q)
get.gamma.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.gamma.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
get.gamma.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.gamma.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
graphics::par(old.par)

---

Fitting parameters of a Gompertz distribution from two or more quantiles

Description

get.gompertz.par returns the parameters of a Gompertz distribution where the pth percentiles match with the quantiles q.

Usage

get.gompertz.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.gompertz.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a Gompertz distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

Comply with a parametrization of this distribution. The definition of this distribution in the literature is not unique.

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pgompertz from the package eha for distribution implementation details.

Examples

q <- eha::qgompertz(p = c(0.025, 0.5, 0.975), shape = 2, scale = 5)
old.par <- graphics::par(mfrow = c(2, 3))
get.gompertz.par(q = q)
get.gompertz.par(q = q, fit.weights = c(100, 1, 100))
get.gompertz.par(q = q, fit.weights = c(10, 1, 10))
get.gompertz.par(q = q, fit.weights = c(1, 100, 1))
get.gompertz.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- eha::qgompertz(p = c(0.025, 0.5, 0.975), shape = 0.2, scale = 0.5)
old.par <- graphics::par(mfrow = c(2, 3))
get.gompertz.par(q = q)
get.gompertz.par(q = q, fit.weights = c(100, 1, 100))
get.gompertz.par(q = q, fit.weights = c(10, 1, 10))
get.gompertz.par(q = q, fit.weights = c(1, 100, 1))
get.gompertz.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- eha::qgompertz(p = c(0.025, 0.5, 0.975), shape = 1, scale = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.gompertz.par(q = q)
get.gompertz.par(q = q, fit.weights = c(100, 1, 100))
get.gompertz.par(q = q, fit.weights = c(10, 1, 10))
get.gompertz.par(q = q, fit.weights = c(1, 100, 1))
get.gompertz.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantiles
q <- eha::qgompertz(p = c(0.025, 0.975), shape = 2, scale = 5)
old.par <- graphics::par(mfrow = c(2, 3))
get.gompertz.par(p = c(0.025, 0.975), q = q)
get.gompertz.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1), scaleX = c(0.0001, 0.9999))
get.gompertz.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
get.gompertz.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.gompertz.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
graphics::par(old.par)

---

Fitting parameters of a hypergeometric distribution from three or more quantiles

Description

get.hyper.par returns the parameters of a hypergeometric distribution where the pth percentiles match with the quantiles q.

Usage

get.hyper.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least three. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.hyper.par uses the R function optim with the method L-BFGS-B.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a hypergeometric distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See phyper for distribution implementation details.

Examples

q <- stats::qhyper(p = c(0.025, 0.5, 0.975), m = 5, n = 3, k = 3)
old.par <- graphics::par(mfrow = c(2, 3))
get.hyper.par(q = q)
get.hyper.par(q = q, tol = 1)
get.hyper.par(q = q, fit.weights = c(100, 1, 100))
get.hyper.par(q = q, fit.weights = c(10, 1, 10))
get.hyper.par(q = q, fit.weights = c(1, 100, 1))
get.hyper.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qhyper(p = c(0.025, 0.5, 0.975), m = 10, n = 5, k = 4)
old.par <- graphics::par(mfrow = c(2, 3))
get.hyper.par(q = q)
get.hyper.par(q = q, fit.weights = c(100, 1, 100))
get.hyper.par(q = q, fit.weights = c(10, 1, 10))
get.hyper.par(q = q, fit.weights = c(1, 100, 1))
get.hyper.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

---

Fitting parameters of a lognormal distribution from two or more quantiles

Description

get.lnorm.par returns the parameters of a lognormal distribution where the pth percentiles match with the quantiles q.

Usage

get.lnorm.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.lnorm.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method Nelder-Mead will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a lognormal distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

Comply with a parametrization of this distribution. The definition of this distribution in the literature is not unique.

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See plnorm for distribution implementation details.

Examples

q <- stats::qlnorm(p = c(0.025, 0.5, 0.975), meanlog = 4, sdlog = 0.8)
old.par <- graphics::par(mfrow = c(2, 3))
get.lnorm.par(q = q)
get.lnorm.par(q = q, fit.weights = c(100, 1, 100))
get.lnorm.par(q = q, fit.weights = c(10, 1, 10))
get.lnorm.par(q = q, fit.weights = c(1, 100, 1))
get.lnorm.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qlnorm(p = c(0.025, 0.5, 0.975), meanlog=-4, sdlog = 0.8)
old.par <- graphics::par(mfrow = c(2, 3))
get.lnorm.par(q = q)
get.lnorm.par(q = q, fit.weights = c(100, 1, 100))
get.lnorm.par(q = q, fit.weights = c(10, 1, 10))
get.lnorm.par(q = q, fit.weights = c(1, 100, 1))
get.lnorm.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qlnorm(p = c(0.025, 0.5, 0.975), meanlog = 1, sdlog = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.lnorm.par(q = q)
get.lnorm.par(q = q, fit.weights = c(100, 1, 100))
get.lnorm.par(q = q, fit.weights = c(10, 1, 10))
get.lnorm.par(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.000001, 0.99999999))
get.lnorm.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qlnorm(p = c(0.025, 0.5, 0.975), meanlog = 0.1, sdlog = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.lnorm.par(q = q)
get.lnorm.par(q = q, fit.weights = c(100, 1, 100))
get.lnorm.par(q = q, fit.weights = c(10, 1, 10))
get.lnorm.par(q = q, fit.weights = c(1, 100, 1))
get.lnorm.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qlnorm(p = c(0.025, 0.975), meanlog = 4, sdlog = 0.8)
old.par <- graphics::par(mfrow = c(2, 3))
get.lnorm.par(p = c(0.025, 0.975), q = q)
get.lnorm.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1), scaleX = c(0.1, 0.001))
get.lnorm.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100), scaleX = c(0.1, 0.001))
get.lnorm.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.lnorm.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
graphics::par(old.par)

---

Fitting parameters of a logistic distribution from two or more quantiles

Description

get.logis.par returns the parameters of a logistic distribution where the pth percentiles match with the quantiles q.

Usage

get.logis.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.logis.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values. The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a logistic distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See plogis for distribution implementation details.

Examples

q <- stats::qlogis(p = c(0.025, 0.5, 0.975), location = 0, scale = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.logis.par(q = q)
get.logis.par(q = q, scaleX = c(0.5, 0.5))
get.logis.par(q = q, fit.weights = c(100, 1, 100))
get.logis.par(q = q, fit.weights = c(10, 1, 10))
get.logis.par(q = q, fit.weights = c(1, 100, 1))
get.logis.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qlogis(p = c(0.025, 0.5, 0.975), location = 0, scale = 3)
old.par <- graphics::par(mfrow = c(2, 3))
get.logis.par(q = q)
get.logis.par(q = q, fit.weights = c(100, 1, 100))
get.logis.par(q = q, fit.weights = c(10, 1, 10))
get.logis.par(q = q, fit.weights = c(1, 100, 1))
get.logis.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qlogis(p = c(0.025, 0.975), location = 0, scale = 3)
old.par <- graphics::par(mfrow = c(1, 3))
get.logis.par(p = c(0.025, 0.975), q = q)
get.logis.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.logis.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
graphics::par(old.par)

---

Fitting parameters of a negative binomial distribution from two or more quantiles

Description

get.nbinom.par returns the parameters of a negative binomial distribution where the pth percentiles match with the quantiles q.

Usage

get.nbinom.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.nbinom.par uses the R function optim with the method L-BFGS-B.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a negative binomial distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pnbinom for distribution implementation details.

Examples

q <- stats::qnbinom(p = c(0.025, 0.5, 0.975), size = 10, prob = 0.5)
old.par <- graphics::par(mfrow = c(2, 3))
get.nbinom.par(q = q)
get.nbinom.par(q = q, fit.weights = c(100, 1, 100))
get.nbinom.par(q = q, fit.weights = c(1, 100, 1))
get.nbinom.par(q = q, fit.weights = c(10, 1, 10))
get.nbinom.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

q <- stats::qnbinom(p = c(0.025, 0.5, 0.975), size = 1, prob = 0.5)
old.par <- graphics::par(mfrow = c(2, 3))
get.nbinom.par(q = q, tol = 0.01)
get.nbinom.par(q = q, fit.weights = c(100, 1, 100))
get.nbinom.par(q = q, fit.weights = c(1, 100, 1), tol = 0.01)
get.nbinom.par(q = q, fit.weights = c(10, 1, 10), tol = 0.01)
get.nbinom.par(q = q, fit.weights = c(1, 10, 1), tol = 0.01)
graphics::par(old.par)

q <- stats::qnbinom(p = c(0.025, 0.5, 0.975), size = 1, prob = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.nbinom.par(q = q)
get.nbinom.par(q = q, fit.weights = c(100, 1, 100))
get.nbinom.par(q = q, fit.weights = c(1, 100, 1))
get.nbinom.par(q = q, fit.weights = c(10, 1, 10))
get.nbinom.par(q = q, fit.weights = c(1, 10, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qnbinom(p = c(0.025, 0.975), size = 10, prob = 0.5)
old.par <- graphics::par(mfrow = c(2, 3))
get.nbinom.par(p = c(0.025, 0.975), q = q,)
get.nbinom.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.nbinom.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
get.nbinom.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.nbinom.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
graphics::par(old.par)

---

Fitting parameters of normal distribution from two or more quantiles

Description

get.norm.par returns the parameters of a normal distribution where the pth percentiles match with the quantiles q.

Usage

get.norm.par(p = c(0.025, 0.5, 0.975), q, show.output = TRUE, 
   plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least two. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.norm.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a normal distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pnorm for distribution implementation details.

Examples

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 12, sd = 34)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.par(q = q)
get.norm.par(q = q, scaleX = c(0.00001, 0.99999))
get.norm.par(q = q, fit.weights = c(10, 1, 10))
get.norm.par(q = q, fit.weights = c(1, 10, 1))
get.norm.par(q = q, fit.weights = c(100, 1, 100))
get.norm.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0, sd = 1)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.par(q = q)
get.norm.par(q = q, fit.weights = c(10, 1, 10))
get.norm.par(q = q, fit.weights = c(1, 10, 1))
get.norm.par(q = q, fit.weights = c(100, 1, 100))
get.norm.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0.1, sd = 0.1)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.par(q = q)
get.norm.par(q = q, fit.weights = c(10, 1, 10))
get.norm.par(q = q, fit.weights = c(1, 10, 1))
get.norm.par(q = q, fit.weights = c(100, 1, 100))
get.norm.par(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

## example with only two quantiles
q <- stats::qnorm(p = c(0.025, 0.975), mean = 12, sd = 34)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.par(p = c(0.025, 0.975), q = q)
get.norm.par(p = c(0.025, 0.975), q = q, fit.weights = c(10, 1))
get.norm.par(p = c(0.025, 0.975), q = q, fit.weights = c(100, 1))
get.norm.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 10))
get.norm.par(p = c(0.025, 0.975), q = q, fit.weights = c(1, 100))
graphics::par(old.par)

---

Fitting standard deviation of a normal distribution from one or more quantiles and known mean

Description

get.norm.sd returns the standard deviation of a normal distribution where the pth percentiles match with the quantiles q.

Usage

get.norm.sd(p = c(0.025, 0.5, 0.975), q, show.output = TRUE, plot = TRUE,
                   fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, vector of probabilities.
q	numeric, vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default vaule is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities and the number of quantiles must be identical and should be at least two. get.norm.sd uses the central limit theorem and the linear regression.

If show.output = TRUE the output of the function lm will be shown.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function will be meaningful only if the quantile comes from a normal distribution.

Value

Returns an estimated standard deviation or missing value

Note

It should be noted that the data must be normally distributed, or the central limt theorem must hold for large (enough) samples sizes.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See pnorm for distribution implementation details.

Examples

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0, sd = 2)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.sd(q = q)
get.norm.sd(q = q, scaleX = c(0.0001, 0.9999))
get.norm.sd(q = q, fit.weights = c(10, 1, 10))
get.norm.sd(q = q, fit.weights = c(1, 10, 1))
get.norm.sd(q = q, fit.weights = c(100, 1, 100))
get.norm.sd(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 176, sd = 15)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.sd(q = q)
get.norm.sd(q = q, fit.weights = c(10, 1, 10))
get.norm.sd(q = q, fit.weights = c(1, 10, 1))
get.norm.sd(q = q, fit.weights = c(100, 1, 100))
get.norm.sd(q = q, fit.weights = c(1, 100, 1))
graphics::par(old.par)

## The estimation model is not suitable for the following quantiles.
## Because the quantile is unsymmetrical, which could not be from a normally distributed data.
q <- c(-2, 30, 31)
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.sd(q = q)
get.norm.sd(q = q, fit.weights = c(10, 1, 10))
get.norm.sd(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.0001, 0.9999))
get.norm.sd(q = q, fit.weights = c(100, 1, 100))
get.norm.sd(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.0001, 0.9999))
graphics::par(old.par)

## Estimating from actually exponentially distributed data
x.exp <- rexp(n = 10, rate = 5)
mean(x.exp)
stats::sd(x.exp)
q <- quantile(x.exp, c(0.025, 0.5, 0.975))
old.par <- graphics::par(mfrow = c(2, 3))
get.norm.sd(q = q)
get.norm.sd(q = q, fit.weights = c(1, 10, 1))
get.norm.sd(q = q, fit.weights = c(10, 1, 10))
get.norm.sd(q = q, fit.weights = c(1, 100, 1))
get.norm.sd(q = q, fit.weights = c(100, 1, 100))
graphics::par(old.par)

## other examples
q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 1, sd = 1)
get.norm.sd(q = q)

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 1, sd = 0.5)
get.norm.sd(q = q)

q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0.01, sd = 0.1)
get.norm.sd(q = q)

---

Fitting parameters of a pert distribution from four or more quantiles

Description

get.pert.par returns the parameters of a pert distribution where the pth percentiles match with the quantiles q.

Usage

get.pert.par(p = c(0.025, 0.5, 0.6, 0.975), q, show.output = TRUE,
    plot = TRUE, tol = 0.001, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...)

Arguments

p	numeric, single value or vector of probabilities.
q	numeric, single value or vector of quantiles corresponding to p.
show.output	logical, if TRUE the optim result will be printed (default value is TRUE).
plot	logical, if TRUE the graphical diagnostics will be plotted (default value is TRUE).
tol	numeric, single positive value giving the absolute convergence tolerance for reaching zero (default value is 0.001).
fit.weights	numerical vector of the same length as a probabilities vector p containing positive values for weighting quantiles. By default all quantiles will be weighted by 1.
scaleX	numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if plot = TRUE). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile.
...	further arguments passed to the functions plot and points (relevant only if plot = TRUE).

Details

The number of probabilities, the number of quantiles and the number of weightings must be identical and should be at least three. Using the default p, the three corresponding quantiles are the 2.5th percentile, the median and the 97.5th percentile, respectively. get.pert.par uses the R function optim with the method L-BFGS-B. If this method fails the optimization method BFGS will be invoked.

If show.output = TRUE the output of the function optim will be shown. The item convergence equal to 0 means the successful completion of the optimization procedure, otherwise it indicates a convergence error. The item value displays the achieved minimal value of the functions that were minimized.

The estimated distribution parameters returned by the function optim are accepted if the achieved value of the minimized function (output component value of optim) is smaller than the argument tol.

The items of the probability vector p should lie between 0 and 1.

The items of the weighting vector fit.weights should be positive values.

The function which will be minimized is defined as a sum of squared differences between the given probabilities and the theoretical probabilities of the specified distribution evaluated at the given quantile points (least squares estimation).

Value

Returns fitted parameters of a pert distribution or missing values (NA's) if the distribution cannot fit the specified quantiles.

Note

It should be noted that there might be deviations between the estimated and the theoretical distribution parameters in certain circumstances. This is because the estimation of the parameters is based on a numerical optimization method and depends strongly on the initial values. In addition, one must always keep in mind that a distribution for different combinations of parameters may look very similar. Therefore, the optimization method cannot always find the "right" distribution, but a "similar" one.

If the function terminates with the error massage "convergence error occurred or specified tolerance not achieved", one may try to set the convergence tolerance to a higher value. It is yet to be noted, that good till very good fits of parameters could only be obtained for tolerance values that are smaller than 0.001.

Author(s)

Matthias Greiner [email protected] (BfR),
Katharina Schueller [email protected] (STAT-UP Statistical Consulting),
Natalia Belgorodski [email protected] (STAT-UP Statistical Consulting)

See Also

See ppert from the package mc2d for distribution implementation details.

Examples

q <- mc2d::qpert(p = c(0.025, 0.5, 0.6, 0.975), min = 0, mode = 3, max = 10, shape = 5)
old.par <- graphics::par(mfrow = c(2, 3))
get.pert.par(q = q)
get.pert.par(q = q, fit.weights = c(100, 1, 1, 100))
get.pert.par(q = q, fit.weights = c(10, 1, 1, 10))
get.pert.par(q = q, fit.weights = c(1, 100, 1, 1))
get.pert.par(q = q, fit.weights = c(1, 10, 1, 1))
graphics::par(old.par)

q <- mc2d::qpert(p = c(0.025, 0.5, 0.6, 0.975), min = 1, mode = 5, max = 10, shape = 4)
old.par <- graphics::par(mfrow = c(2, 3))
get.pert.par(q = q)
get.pert.par(q = q, scaleX = c(0.0001, 0.999999))
get.pert.par(q = q, fit.weights = c(100, 1, 1, 100))
get.pert.par(q = q, fit.weights = c(10, 1, 1, 10))
get.pert.par(q = q, fit.weights = c(1, 100, 1, 1))
get.pert.par(q = q, fit.weights = c(1, 10, 1, 1))
graphics::par(old.par)

q <- mc2d::qpert(p = c(0.025, 0.5, 0.6, 0.975), min=-10, mode = 5, max = 10, shape = 4)
old.par <- graphics::par(mfrow = c(2, 3))
get.pert.par(q = q)
get.pert.par(q = q, fit.weights = c(100, 1, 1, 100))
get.pert.par(q = q, fit.weights = c(10, 1, 1, 10))
get.pert.par(q = q, fit.weights = c(1, 100, 1, 1))
get.pert.par(q = q, fit.weights = c(1, 10, 1, 1))
graphics::par(old.par)

q <- mc2d::qpert(p = c(0.025, 0.5, 0.6, 0.975), min=-10, mode = 5, max = 10, shape = 0.4)
old.par <- graphics::par(mfrow = c(2, 3))
get.pert.par(q = q)
get.pert.par(q = q, fit.weights = c(100, 1, 1, 100))
get.pert.par(q = q, fit.weights = c(10, 1, 1, 10))
get.pert.par(q = q, fit.weights = c(1, 100, 1, 1))
get.pert.par(q = q, fit.weights = c(1, 10, 1, 1))
graphics::par(old.par)

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