trapezoid: The Trapezoidal Distribution

In trapezoid: The Trapezoidal Distribution

The Trapezoidal Distribution

Description

Density function, distribution function, quantile function, and random generation for the trapezoidal distribution with minimum equal to min, lower mode equal to mode1, upper mode equal to mode2, and maximum equal to max. For the generalized trapezoidal distribution, n1, n3, and alpha may optionally be specified.

Usage

	dtrapezoid(x, min = 0, mode1 = 1/3, mode2 = 2/3, max = 1, n1 = 2, n3 = 2, 
	  alpha = 1, log = FALSE)
	ptrapezoid(q, min = 0, mode1 = 1/3, mode2 = 2/3, max = 1, n1 = 2, n3 = 2, 
	  alpha = 1, lower.tail = TRUE, log.p = FALSE)
	qtrapezoid(p, min = 0, mode1 = 1/3, mode2 = 2/3, max = 1, n1 = 2, n3 = 2, 
	  alpha = 1, lower.tail = TRUE, log.p = FALSE)
	rtrapezoid(n, min = 0, mode1 = 1/3, mode2 = 2/3, max = 1, n1 = 2, n3 = 2, 
	  alpha = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
min	vector of minima.
mode1	vector of lower modes.
mode2	vector of upper modes.
max	vector of maxima.
n1	vector of growth parameters.
n3	vector of decay parameters.
alpha	vector of boundary ratio parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Details

The generalized trapezoidal distribution is described by van Dorp and Kotz (2003) and van Dorp and colleagues (2007). With n1, n3, and alpha equal to the default values of 2, 2, and 1, respectively, the distribution shape is of a quadrilateral trapezoid. Altering n1, n3, or alpha changes the growth rate, decay rate, and boundary ratio parameters, respectively, as demonstrated in the examples below.

Value

dtrapezoid gives the density function, ptrapezoid gives the distribution function, qtrapezoid gives the quantile function, and rtrapezoid generates random deviates.

Author(s)

Jeremy Thoms Hetzel [e-mail]

References

van Dorp, J. R. and Kotz, S. (2003) Generalized trapezoidal distributions. Metrika. 58(1):85–97. Preprint available: http://www.seas.gwu.edu/~dorpjr/Publications/JournalPapers/Metrika2003VanDorp.pdf

van Dorp, J. R., Rambaud, S.C., Perez, J. G., and Pleguezuelo, R. H. (2007) An elicitation procedure for the generalized trapezoidal distribution with a uniform central stage. Decision Analysis Journal. 4:156–166. Preprint available: http://www.seas.gwu.edu/~dorpjr/Publications/JournalPapers/DA2007.pdf

See Also

Distributions for standard distributions.

Examples

	## Plot default trapezoid distribution
	curve(dtrapezoid(x, min = 0, mode1 = 1/3, mode2 = 2/3, max = 1, 
	  n1 = 2, n3 = 2, alpha = 1), from = 0, to = 1)
	
	
	## Plot triangular trapezoid distribution
	curve(dtrapezoid(x, min = 0, mode1 = 1/2, mode2 = 1/2, max = 1, 
	  n1 = 2, n3 = 2, alpha = 1), from = 0, to = 1)
	
	
	## Explore effects of n1, n3, and alpha parameters
	# plyr and ggplot2 are required for this example
	require(plyr)
	require(ggplot2)
	
	x <- seq(from = 0, to = 1, by = 0.01)	
	
	# Create a list of arguments, varying n1, n3, and alpha
	arguments <- list()
	arguments[['A']] <- list(x = x, n1 = 2, n3 = 2, alpha = 0.8)
	arguments[['B']] <- list(x = x, n1 = 1.5, n3 = 1.5, alpha = 1)
	arguments[['C']] <- list(x = x, n1 = 2.5, n3 = 2.5, alpha = 1.5)
	arguments[['D']] <- list(x = x, n1 = 1.5, n3 = 2.5, alpha = 0.5)
	arguments[['E']] <- list(x = x, n1 = 2.5, n3 = 1.5, alpha = 1)
	arguments[['F']] <- list(x = x, n1 = 0.5, n3 = 0.5, alpha = 1.5)
	arguments[['G']] <- list(x = x, n1 = 1.5, n3 = 0.5, alpha = 0.5)
	arguments[['H']] <- list(x = x, n1 = 2.5, n3 = 0.5, alpha = 1)
	arguments[['I']] <- list(x = x, n1 = 0.5, n3 = 1.5, alpha = 1.5)
	arguments[['J']] <- list(x = x, n1 = 0.5, n3 = 2.5, alpha = 0.5)
	
	# Calculate the distributions
	plot.data <- ldply(arguments, function(z)
	{
	  x <- z$x
	  density <- dtrapezoid(x = z$x, min = 0, mode1 = 0.2, mode2 = 0.8,
	    max = 1, n1 = z$n1, n3 = z$n3, alpha = z$alpha)
	  args <- paste("n1 = ", z$n1, ", n3 = ", z$n3, ", alpha = ", z$alpha, 
	    sep="", collapse="")
	  out <- data.frame(x, density, args)
	})
	
	# Create labels for later use in displaying the arguments on the plots 
	plot.data$label <- paste(plot.data$.id, ": ", plot.data$args, sep="")
	
	# Create plots
	generalizedTrapezoids <- ggplot(data = plot.data, aes(x = x, y = density)) + 
	  geom_line() + theme_bw() + 
	  facet_wrap(~label, ncol = 2, scales = "free_y")
	print(generalizedTrapezoids)

trapezoid documentation built on June 9, 2022, 5:08 p.m.